text:
\section{Introduction}
Broader participation by diverse demographic and geographic groups is key to building strength in STEM fields in general, and High Energy Physics (HEP) is no exception. A recent report from the National Science Board Vision 2030~\cite{NSF2020_vision} urges that faster progress in increasing diversity is needed to reduce the significant talent gap in the United States (U.S.). It also indicates that foreign-born individuals are significant contributors to U.S. Science and Engineering. The data shows that a global science and engineering enterprise is growing faster than before, while the U.S. share of scientific discovery is dropping. The silver lining is that the U.S. has a significant lead in fundamental research, and we must innovate and embrace ways to be more inclusive and diverse to maintain our global leadership. For U.S. HEP to stay competitive and evolve stronger, we must address these issues as part of the planning process for the next decade so that an influx of new workforce in the field can be encouraged, retaining those who are in the field and as well equip them with a skills to find careers both within and outside HEP.

The current trend of education, diversity, gender and career opportunities in HEP~\cite{ICHEP_div} is highly selective, and generally exclusionary of participation by underrepresented minority and female students. To address this issue, meaningful opportunities for broader participation are required at different levels. While a proposed Master's Degree in Applied Physics could enable a technological career in laboratories, accelerator facilities, private sector, or possibly a PhD degree, it can also attract students from less favored communities who may wish to transition to a career with a Masters degree.

At Ph.D. and post-doctoral level, a vast majority of researchers who choose to stay in academia are hired as teaching faculty at  Predominantly Undergraduate Institutions  (PUIs)~\cite{pui, pui_list} and Community Colleges (CCs). Moreover, these institutions host a vast majority of students from underrepresented groups. The faculty of these institutions, despite their talents, are at disadvantage when it comes to continuity in research that requires commitment of both time and resources. Typically, higher teaching loads, lack of informed guidance,and lack of collaborative opportunities diminish their chances for being competitive for research funding.

 The strength of the particle physics community lies in its openness, diversity, and access to some of the best creative talent worldwide. This strength has resulted in the advancement of human knowledge that is leading to a detailed understanding of the inner workings of the universe. Engaging undergraduate institutions and their faculty and students in this front-line research enterprise would both enrich and strengthen the workforce and the field.

With the advent of tools for Open Science, we are poised for an even greater and more significant role, and in a more inclusive way. For example, CERN Open Data~\cite{cern_opendata} is being harnessed to connect discovery science with the public through outreach activities, techniques such as Machine Learning, and much more.
Development of state-of-the-art particle sensors often find applications outside of particle physics, e.g. in medical imaging, material diagnostics, etc. Small scale development and application research is well in reach of smaller research groups with limited resources, if they can be teamed up with larger scale HEP collaborations.
The software training in the HEP community is transforming the preparation of our next generation of problem solvers by reaching students beyond HEP.

We must strengthen existing successful methodologies, many of which have been created and developed by the U.S. HEP community, and explore new ways of transferring HEP resources, talent, and data into an Open Science paradigm. By doing so we propel our field forward, and garner the interest and support of general public through the next phase of scientific discoveries.
Simultaneously, we must prepare HEP Ph.D. students  and postdocs for careers inside or outside academia.

We recognize that advancements and discoveries in physics have become a huge enterprise necessitating advances in computing, software, AI, Machine Learning and detector technologies in order to develop, design, build and operate experiments.   And these consume a significant fraction of time of physicists. This means that our perception of what it means to 'be a physicist' and what work is considered a contribution to the physics must evolve and expand. In the following sections we discuss several proposals to address and mitigate the aforementioned challenges.

\section{Assessing the needs of Particle Physics}

The Standard Model of particle physics provides an an impressive description of much of observed particle physics. Nevertheless, it has significant shortcomings and cannot be the ultimate theory of nature at the fundamental scale. Theoretical particle physics provides a plethora of extensions and modifications of the Standard Model, many of which predict new physics phenomena for which there are either minimal or no observational motivations at present. Further groundbreaking experimental programs will be required to make further scientific discoveries.  These can be forged by going to higher energy, measuring more precisely or by performing measurements with new techniques.  It is not unreasonable to assume that new discoveries will incorporate all three of these attributes.

In all cases the progress will be technologically driven, in accelerator technology, in sensor technology, or in data treatment and analysis technology.
Strong injection of technological know-how is therefore a requirement for progress. Much of the technological; progress does not necessarily require extensive particle physics knowledge. Rather it does require a deep understanding of technology and the ability to innovate and create new technology.

Unlocking the technological work-force needed for future particle physics research requires recruiting talent outside the normal particle physics oriented study and career paths.
There is potentially large pool of students and faculties at PUIs and CCs that can be tapped to increase such participation. The faculty could contribute to HEP along with students who would acquire various skills that prepares them for STEM careers.

That this potential is not well accessed at the moment is due to resource limitations in these faculties, in terms of research time and equipment and travel budgets. There is also little awareness of the potential for significant contribution from these faculties with the faculties themselves and with the current main players in HEP, the R1~\cite{pui} universities and research institutions.

In sections below, suggestions are made to implement programs to unlock the potential for technical education and research connected to experimental particle physics.  While such a programs would contribute to the development of new workforce to meet the technical challenges of experimental particle physics, they would also contribute to making the HEP community more diverse and offer new opportunities for participation from those underrepresented in STEM. The next two decades of the Snowmass2021 vision can afford an opportunity to review the challenges and develop strategies to correct the disparities and transform the particle physics field into a stronger and more diverse ecosystem of talent, expertise and public-minded support. Working together across the various groups of the Community Engagement Frontier and in consort with the scientific, technical and computation frontiers of the Snowmass2021 Community Self-study, the community can develop an understanding and project the person-power needs of the field in terms of students, physicists, and technical, analytical, and managerial eperts on an annual basis – and in parallel – project the societal benefits that could be acquired through this process.

Among the facets of this proposed study:
\begin{itemize}
 \item Needs Assessment of the Community: Work with the various research and technical
frontiers to develop projections of their person-power and educational needs on an
annual basis over the next two decades.
 \item Needs Assessment of the Education Process: Review of teaching at all levels –
particularly transition points between education levels – where "transmission
coefficients" of students are low and develop ideas to improve these circumstances.
Understanding these are particularly important to increasing the participation of
underrepresented groups in the field.
 \item Model Development to Guide Community Planning: Engage with AIP and APS who
follow the education process of younger students upward through schools to develop
statistical models of what currently is the situation and how it might be improved, not
only to the benefit of particle physics, but also the broader STEM community.
\end{itemize}

\section{Faculty Collaboration across Academia}

The field of Particle Physics has successfully brought many young researchers through the post-doctoral ranks, at which time these individuals are then searching for more permanent positions in academia, National Laboratories, and in the private sector. In the domain of academia, many individuals may consider opportunities at Carnegie classified R1 universities, other institutions including undergraduate-serving colleges and universities, as well as community colleges.

For those seeking what are typically teaching-rich positions, the potential exists for these persons  to attract a broader geographic and demographic base of students to the particle physics field which affords the possibility of strengthening the participation of underrepresented groups in the particle physics research program. Engaging those teaching at undergraduate institutions in front-line particle physics research will strengthen our academic workforce, and in turn inspire their students.

Recommendations:
\begin{itemize}
\item Conduct an analysis to assess the needs of faculty at undergraduate institutions in order for them to be successful in developing vibrant research programs, whilst maintaining a significant teaching load characteristic of such institutions.
\item Survey institutional collaborations of PUI and CC faculty with R1 and laboratory groups that have proven successful so far, to assess lessons learned.
\item Conduct a study of new models of collaboration or cooperation that would allow PUI/CC faculty and their students to collaborate in demonstrably effective ways in experiments, and for faculty to be effective leader rather than regarded as secondary.
\item Conduct a survey of R1 institutions and research laboratory physicists who might share an interest in collaborating with PUI and CC faculty.
\item Identify and inventory past and present activities by PUI and CC faculty engaged in HEP or related activities such as
experiment design, construction, installation, and maintenance.
\end{itemize}

\section{New Masters Degree in Applied Physics}

 A Master's Degree in Applied Physics is proposed here with the aim of providing an advanced degree beyond bachelors level that could open meaningful career paths for students into technological careers in laboratories or the private sector; additionally such a degree could serve as a springboard for those who find that a PhD degree is possible.  This would provide a meaningful opportunity for participation by underrepresented groups and enrich and broaden participation in STEM careers.

Recommendations:
\begin{itemize}
\item Conduct a needs analysis to assess the value and chances of success of such a program.
\item Compare such a program with current models of Masters Degrees in engineering and MBA degrees.
\item Conduct an assessment of private sector buy in and support for such a degree opportunity for current students and for their employees seeking career improvement.
\item Assess the effectiveness of such a degree to  provide training opportunities for those seeking technical careers at laboratories in our field or elsewhere or perhaps medical fields.
\item Identify the curriculum for such degree(s) and whether they could be made available nationwide through shared curriculum.
\item Assess university, PUI and CC buy in and participation in such a program.
\end{itemize}

\section{Training the HEP community}

Software, computing, hardware and analysis skills are an integral part of the toolkit of successful HEP experimentalists, and maximizing the science from the investments in current and future HEP projects relies critically on it. This is also a skill set that is readily transferable in case of non-HEP career evolution of people trained in HEP. For example, software training is now key in many research fields, but most users learn software skills only after joining a research program. Individual universities do not uniformly provide such training to students prior to their beginning their Ph.D. research. HEP Computing and hardware skills at university level are even more rare given significant costs and time investments. In addition, domain-specific aspects add challenges to the learning process. Embarking on a HEP-specific training could still involve experiment-specific  environment challenges. Figure~\ref{fig:pyramid} shows the HEP related education and research training evolution from a student to faculty level and with examples of  training and  programs  indicated at different levels. It is important to point out that we must find ways to make sure that diversity, equity and inclusiveness percolates all the way upwards from the bottom to the pyramid of Figure~\ref{fig:pyramid}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=35pc]{Pyramid.png}\hspace{5pc}%
\begin{minipage}[b]{28pc}\caption{\label{fig:pyramid}Evolution of HEP Education and Training}
\end{minipage}
\end{center}
\end{figure}

Some of the current training programs in the field of HEP are described below. It would be very beneficial if some of such training could accorded course credit, even if taken during the summer, so that not only highly competitive students but a bigger subset of students is encouraged and able to take part. And the scientists, faculty, and postdocs involved engaged in these training's should be given compensation for their significant time investment.

\subsection{Software Training}
No one size fits all while imparting software training. A possible HEP wide solution is to exploit our large-scale community structure and organize training within our research domains~\cite{training_springer}. Recent successful and ongoing efforts like IRIS-HEP~\cite{iris_hep}, FIRST-HEP~\cite{first_hep} and HEP Software Foundation (HSF)~\cite{hsf_hep}  have taken strong and effective steps in this direction and impart training to those within the field and ancillary fields. The HEP community must play a role in bringing together a focus on these efforts to engender sustainability and scalability.  It is important to initiate learning early on in the process of preparing a software-equipped future particle physicist.

At PUIs and CCs, and often at R1 universities, students pursuing undergraduate degrees in physics do not necessarily get exposed to all the core undergraduate physics and computing courses that are related with research. When going on to go graduate school, these students may lag behind their peers. While the curricular issue of what core physics courses are taught at student's institution may be challenging to address, nevertheless we can and should develop a way for students to take elective Particle Physics courses that give them university credit, and in addition provide HEP training in related software. While students may exit at different levels in the HEP education and training pyramid of Figure~\ref{fig:pyramid}, we should try to make sure that opportunities for training reach all.

The software training programs of IRIS-HEP and HSF have trained over 1500 undergrads, Masters, PhDs, postdocs and senior physicists with software skills over the last 3 years, even in times of COVID-19 challenges. One of the commendable achievements of software training has been the building of community around LHC experiments~\cite{training_lhcexp} and DUNE~\cite{training_dune}. Nuclear Physics, which has similar educational needs, is also engaged with their effort led by Jefferson Lab (TJNAF).

\subsection{Training Internships}

Internship programs by DOE~\cite{needs_doeintern} and NSF/REU~\cite{needs_nsfreu} have been very popular to encourage students to gain valuable research experience and are highly competitive.  For example DOE sponsored internships like SULI~\cite{needs_suli} place undergraduate physics or engineering majors in paid  summer internships. These internships~\cite{needs_fnalintern} offer a chance for students and professionals (school teachers and visiting faculty) to work with scientists or engineers on projects at the frontier of scientific research in particle physics.
For all students, but particularly those from academic departments that are primarily PUI or masters degree oriented and which often attract community college transfers, the opportunities afforded by extended visits to national laboratories are truly life changing.  Efforts by Wichita State University to attract and train students for STEM careers is an excellent example of such efforts on behalf students by dedicated and creative faculty.

\subsection{Experiment Training Efforts}

Several experiments have their own unique programs to train new collaborators  - PhDs, postdocs or faculty in all aspects of experiment. The CMS Experiment~\cite{training_cms,training_cmsdas} and its contingent of US physicists (USCMS) that maintains the Fermilab LHC Physics Center (LPC)~\cite{training_lpcfnal} and an independent LHCb Experiment program~\cite{training_lhcb} are examples of very successful stories. The LPC has recently started a Summer Undergraduate Research Internship program to train students on HEP specific topics like detector hardware, physics analysis and software. Through such a program, students interested in HEP can be trained early on and the skills are translatable to other STEM areas, should the students choose to go in other career directions.

It is notable that the R\&D and construction phase of a typical HEP experimental project can span several years and often a decade or more, and affords a fertile opportunity to learn detector hardware related to HEP.  And, while major work happens at the laboratory or hosting site of the experiment, substantial building and testing of detector components can be distributed across the collaborating institutions,  This can be broadly beneficial to faculty and students across a wide range of participating colleges, universities and laboratories.

Ongoing construction of the CMS Phase-2 Pixel sub-detector~\cite{training_cornellintern,training_cornellpixel} for the CMS experiment at Cornell University and sub-detectors for MINOS~\cite{training_minos}, NOvA~\cite{training_nova} and Mu2e~\cite{training_mu2e} experiments at Minnesota are some of the very good examples that involve hundreds of students over the course of construction from different STEM backgrounds at the host and collaborating  institutions. Figure~\ref{fig:nova} shows the spectrum of undergraduates and their major fields of study, who were involved in the building of the NOvA detector~\cite{training_novavideo}. That experience demonstrated that undergrads can be actively and effectively engaged, alert to and fully capable of recognizing anomalies and solving problems. The Students and postdocs appreciated being supported  as valued team members in the detector building tasks, and for the hands on experience of how to work collaboratively and manage the work of the various teams. Graduate students and postdocs learned about the challenges of building a detector, managing others, and paying attention to time and budget. Moreover, this type of experience is compelling to a general audience and newsworthy, as one reporter captured, "I worked in a coffee shop--this is quite different. I wanted to work in an environment where I could live and breathe science and to be around people who've had similar experiences and classes."

\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth, frame]{nova.png}\hspace{5pc}%
\begin{minipage}[b]{28pc}\caption{\label{fig:nova}Over 700 undergraduate students from different disciplines built NO$v$A Detector at University of Minnesota}
\end{minipage}
\end{center}
\end{figure}

\subsection{Undergraduate Research at PUI's} Initiating meaningful opportunities for research early in a student's career can often lead to the life and career-altering realization that a student actually has the capacity to be a scientist. That all-important self-identification can be the result of well coordinated programs with committed faculty mentors, who are teamed with graduate students and postdocs.  And access to suitable facilities is also a signature of an effective program.  Features of many research invested PUI's are the commitment of a cadre of mentors to attract students to the excitement of the science and attendant research opportunities, and notably, and the participation of engaged students, the most accomplished among them as good as any University's best. As access to research grade facilities may be more challenging for PUIs, we support proposals to develop PUI-centric undergraduate research collaborations, especially in EPSCoR states~\cite{EPSCoR_DOE}~\cite{EPSCoR_NSF}, where program leadership and coordination of HEP related research projects evolve to capitalize on the diversity of talent at the nation's smaller and likely more rural universities and colleges.

\section{Open Science to build HEP community}

The process of sharing scientific findings in journals has been the mode of Open Science~\cite{os_forum} for many decades~\cite{os_broadaccess}, and  Particle Physics and Astronomy were among the first fields to embrace this approach, fostering and sharing open source software, open access to published work and open data for researchers and citizen scientists. The World Wide Web, Invenio, Indico, Inspire, Zenodo and  OSG (Open Science Grid) are examples of some of the platforms and tools provided by the domain of particle physics to the scientific community and beyond. CERN~\cite{os_access} provides access to open source hardware, open publishing and open data via the CERN Open Hardware License, the Sponsoring Consortium for Open Access Publishing in Particle Physics (SCOAP3~\cite{os_physics})and the Open Data Portal for the LHC experiments. The strength of the particle physics HEP community lies in its openness, diversity and access to some of the best creative talent worldwide. With the advent of tools to operate Open Science, we are poised for an even greater and more significant role, and in a more inclusive way.

\subsection{Benefits}

Open Science empowers the ability to collaborate, question and contribute to sustainable scientific knowledge and process, and accelerate future discoveries. Open data from LHC~\cite{cern_opendata} is employed by educators, students, and self-learners for teaching, learning, and research. It allows research to maximize its impact by reaching a wider audience of participation and sharing the findings, seeking collaborations without borders, and expanding funding and career advancement opportunities. It is also greatly enhances the quality and opportunities for education by making it more affordable and equitable.

\subsection{Challenges}

With opportunity comes responsibility. The particle physics community has become fully aware, and needs to make the outside world equally aware, that there are practical limits to making all raw data public.  Besides problems of facilitating training the public at large to analyze data, significant additional resources are needed to support personnel to maintain the data and to serve as consultants as outsiders attempt to use the associated software tools for analysis, which are not typically static over experimental lifetimes, and evolve as experimentation proceeds.  Existing computing infrastructure would not be able to support many others to access all of the available raw data.

Recommendation:

\begin{itemize}
\item The HEP community should define, with cogent arguments, what should be the scope of making our data and resources publicly available, and the hardware, software and person-power costs associated with such implementation.
\end{itemize}

\subsection{Goals}

We must strengthen existing successful methodologies, many of which have been created and developed by the U.S. HEP Community, and explore new ways of translating HEP resources, talent and data into Open Science.
We can advance this powerful framework to create more inclusiveness by Open Software Training and Open Physics teaching that will propel our field and the general public into the next phase of discoveries.

\section{Intersection of Physics and Machine Learning}

Machine Learning (ML) is becoming an integral part of physics research. Many critical HEP algorithms for triggering, reconstruction, and analysis rely on ML and there are entire conferences and summer schools dedicated to this crosscutting field. Moreover, the unique constraints of physics experiments and the ability to exploit symmetries inherent in physics data have made the field of physics-informed ML a vibrant sub-field of Computer Science (CS) research. However, despite the relevance and importance of this research, pursuing a career at the intersection of these fields remains tenuous and undefined endeavor. We must  work towards finding ways about how we can better support academic and career development at the intersection of physics and ML, and ensure that we can continue to benefit from and contribute to state-of-the-art ML.

\subsection{Current Environment}

HEP research has long required advanced, cutting-edge computing techniques, and physicists have historically contributed to the development of these methods. Over the past decades, there have been great advances in the data-processing power of ML algorithms and these methods have been quickly adopted by physicists to address the unique timing, memory, and latency constraints of HEP experiments~\cite{ml_latency}.

Physics is also impacting ML research. The constraints of HEP experiments and known symmetries of physical systems create a rich environment for the development of novel ML (see for example~\cite{ml_neural}). There are even entire conferences and workshops dedicated to this intersection including the Microsoft Physics ML lecture series~\cite{ml_microsoft} and the ML and the Physical Sciences workshop at NeurIPS~\cite{ml_workshop}.

However, despite the demonstrated criticality and vibrancy of this research, from a physics perspective the path to a sustainable research career at the intersection of physics and ML is unclear at best. In fact, many researchers seem to regard ML research and applications as minor side projects\footnote{Due to the lack of community statistics on this topic, the evidence presented here is primarily anecdotal}.

There are very few physics department supported courses on ML\footnote{although~\cite{ml_forphy} provides a good example of how such a course might work} and interested students are often forced to convince CS departments to allow them to attend graduate courses (which typically do not discuss physics-informed ML). Furthermore, at many universities the lack of collaborative interactions between Physics and CS departments limits opportunities for physicists to engage with cutting-edge ML research and develop partnerships with industry researchers\footnote{NYU~\cite{ml_nyu} and UW~\cite{ml_washington} Physics collaborations with the Center for Data Science is an excellent example of cross-department collaboration}.

As a physics graduate student at a US institute it is essentially impossible to focus one's thesis research on ML, as most departments do not acknowledge this as 'physics work' (despite the fact that these algorithms are in many cases critical to the functioning of physics experiments and data processing). Perhaps most egregiously, although there is now some support for this work at the post-doc or research staff level (see IRIS-HEP~\cite{iris_hep} and IAIFI~\cite{ml_iaifi}), early-career researchers are often actively advised against applying for such positions as they constitute 'career death sentences'.

\subsection{Proposals for Future Work}

It is clear that these issues must be addressed in order to best support and advance our field and our researchers. An important first step is to work towards a cultural shift around the perception of what it means to 'be a physicist' and what work is considered and accepted as valid as contribution to physics. We must value technical and software contributions in the same manner we value analysis work and foster a supportive and educational environment for graduate students and early-career researchers interested in ML and physics. It benefits no one to discourage individuals from pursuing work in this vibrant and promising field.

Concretely, this necessitates developing cross-department courses, workshops,and research collaborations. Graduate programs should approach technical and computing skills with the same rigor they do traditional physics courses. Departments and advisors should be encouraged to allow students and post-docs to work across disciplines and explore opportunities for industrial partnerships or internships. This process should also include a reevaluation of requirements for physics graduate theses.

The research landscapes of both physics and ML are shifting, and by adjusting to these changes we can ensure a vibrant future for data-intensive physics and physics-informed ML.

\section{Conclusions}

This paper addresses current needs in education and training in High Energy Physics and how the field can be proactive and innovative in transforming current efforts, which tend to be highly selective and generally exclusionary in nature, into a more broadly participatory and inclusive approach, to the benefit of all students and particularly those from underrepresented demographic groups.  There are significant pools of students and faculty across academia and notably those at Predominantly Undergraduate Institutions and Community Colleges that can be tapped for participation, to the benefit of a stronger and more vibrant HEP field for the future. We can develop this untapped talent by providing creative and innovative opportunities in HEP specific education at undergraduate and Masters levels, and by developing rewarding research opportunities through improved training in physics, instrumentation R\&D and software and analytical techniques. Such efforts will not only ensure that the HEP field has the professional workforce needed to meet the scientific challenges and discoveries that lie ahead, but also the capability to prepare students effectively for productive careers in other STEM fields and across the needs of our broader society.

\section*{References}

\typeout{}


text:
\section{Introduction}

The spectrum of perturbations on large scales is a key test of any
models of the early universe.  During an inflationary era quantum
fluctuations on small scales become stretched beyond the horizon
generating inhomogeneities on super-horizon scales that are otherwise
inexplicable in the standard big bang model~\cite{gw,deltaphi,BST}.
Conventional models of slow-roll inflation generally predict an almost
scale-invariant spectrum of adiabatic density
perturbations~\cite{LL93}.  As major observational programs are now
underway to produce detailed maps of these perturbations, it is
important to investigate whether one can uniquely reconstruct the
inflationary history of the universe from the spectrum of
inhomogeneities.  This question has received considerable attention in
recent years in the context of slow-roll inflation~\cite{recon}
where it has been realised that there is a degeneracy in the spectrum
of adiabatic density perturbations.  This could be removed by a
detection of the gravitational wave background on the same scale
which, in the slow-roll approximation, gives a direct record of the
evolution of the scale factor, and hence of the inflaton potential.

By contrast there has been relatively little study of the
reconstruction of the evolution in the non-slow-roll
case~\cite{nonSR}. Recently it has been discovered that the spectra of
perturbations produced in so-called pre-big bang models of the early
universe~\cite{pbb}, based on solutions of the low energy string
effective action, are invariant under SL(2,R) symmetry transformations
(including S-duality transformations) of the background
fields~\cite{CEW97}. This raises the interesting question of what is
the most general type of cosmological evolution that yields a given
perturbation spectrum.

I will consider linear perturbations, $\delta\phi(\eta,x^i)$,
about a homogeneous background, $\phi(\eta)$, in a homogeneous cosmology.
For a minimally coupled massless field we can neglect any back-reaction
upon the spacetime curvature to first order, and so perturbations obey the
wave equation
\begin{equation}
\label{eom}
\ddot{\delta\phi} + 3H\dot{\delta\phi} - \nabla^2\delta\phi = 0
\ ,
\end{equation}
where a dot denotes derivatives with respect to cosmic time $t$, and
$3H$ corresponds to the expansion rate of the homogeneous
hypersurfaces.
One can decompose the perturbations into independent wavemodes
$\delta\phi_k(\eta)Q_k(x^i)$, where $Q_k(x^i)$ is the eigenfunction
of the spatial Laplacian $\nabla^2$ with eigenvalue $-k^2$.

The canonically normalised quantum field corresponds to the
conformal field perturbation $u=a\delta\phi$, where the scale factor
$a=\int Hdt$.  Perturbations obey the wave equation
\begin{equation}
\label{ukeom}
u_k'' + ( k^2 + \mu^2 ) u_k = 0
 \ ,
\end{equation}
which corresponds to an oscillator with a time dependent effective
mass-squared
\[
\mu^2= - {a'' \over a} \ ,
\]
where a prime denotes a
derivative with respect to conformal time $\eta=\int dt/a$.
During a conventional inflationary era $\mu^2$ is negative and decreases
monotonically, leading to the amplification of vacuum fluctuations.
Modes on arbitrarily small scales ($k^2/|\mu^2|\to\infty$) are
presumed to occupy the flat spacetime vacuum state ($u_k\to
e^{-ik\eta}/\sqrt{2k}$).
These vacuum fluctuations eventually lead to a specific spectrum
of perturbations on large scales ($k^2/\mu^2\to0$), the form of
which is determined solely by $\mu^2(\eta)$.

Consider the most commonly studied case of a power-law
expansion~\cite{PLI}, where the scale factor grows as $a\propto t^p$,
which corresponds, in terms of the conformal time, to
\begin{equation}
\label{pla}
a = a_0 \left( {\eta \over \eta_0} \right)^{(1-2\nu)/2} \ ,
\end{equation}
where
\begin{equation}
\nu = {3\over2} + {1\over p-1} \ .
\end{equation}
Note that $a\to\infty$ as $t\to\infty$ only coincides with the limit
$\eta\to\infty$ for $p<1$. During an inflationary expansion with $p>1$
there is an event horizon and $\eta\to0$ from below as $a\to\infty$.

The effective mass in Eq.~(\ref{ukeom}) is
\begin{equation}
\mu^2=-{\nu^2-1/4 \over \eta^2} \ .
\end{equation}
Note that for $\nu=\pm1/2$ the effective mass $\mu^2$ vanishes and
there is no particle production. This corresponds to a static universe
($p=0$) or a spatially flat Friedmann-Robertson-Walker (FRW) radiation
dominated universe ($p=1/2$).
During an inflationary expansion with $p>1$ and $\nu>3/2$,
$\mu^2\to-\infty$ as $\eta\to0^-$ which leads to fluctuations on
scales larger than the horizon, $|k\eta|>1$.
However, it has previously been noted~\cite{deflation} that a
collapsing universe could in principle produce large scale
perturbations from small scale quantum fluctuations. $|\eta|$ also
represents an effective event horizon in a collapsing model with $p<1$
where $\eta\to0$ from below as $a\to0$.
Any given comoving mode $k$ gets pushed outside horizon
as $|\eta|$ decreases and $k\eta\to0$ as $t\to0^-$ for $p<1$.
For $|\nu|<1/2$, i.e., $0<p<1/2$, the effective mass-squared $\mu^2$ is
positive and becomes large as $\eta\to0^-$ which strongly suppresses
fluctuations on large scales ($|k\eta|\ll1$).

The general solution of the wave equation~(\ref{ukeom}) during a
power-law expansion/contraction is~\cite{LS}
\begin{equation}
u_k(\eta) = \sqrt{|k\eta|} \left[ u_+ H_{|\nu|}^{(1)}(|k\eta|)
+ u_- H_{|\nu|}^{(2)}(|k\eta|) \right] \ .
\end{equation}
where $H^{(i)}_{|\nu|}$ are Hankel functions of order $|\nu|$.
Choosing the quantum vacuum state at early times on small scales
($k\eta\to-\infty$) then determines the spectrum of perturbations
on large scales ($k\eta\to0$)
\begin{equation}
\label{Pu}
{\cal P}_u = {C^2(|\nu|) k^2 (-k\eta)^{1-2|\nu|} \over (2\pi)^2} \ ,
\end{equation}
where the power spectrum is conventionally
defined as ${\cal P}_u=k^3|u_k^2|/2\pi^2$, and the numerical coefficient
\begin{equation}
C(|\nu|) \equiv {2^{|\nu|}\Gamma(|\nu|) \over 2^{3/2}\Gamma(3/2)} \ .
\end{equation}
The spectrum of scalar field perturbations produced on large scales
($|k\eta|\gg1$) can therefore be written as
\begin{equation}
\label{Pdphi}
{\cal P}_{\delta\phi}
= \left( {C(|\nu|) \over \nu-1/2} \right)^2
 \left( {H \over 2\pi} \right)^2 (-k\eta)^{3-2|\nu|} \ .
\end{equation}
In the limit of de Sitter expansion in flat FRW, $p\to\infty$ and
$\nu\to3/2$, we recover the famous result ${\cal
P}_{\delta\phi}=(H/2\pi)^2$ at horizon crossing ($k\eta=-1$) and the
spectrum is independent of scale.

Notice, however, that the spectrum ${\cal P}_u$ given in
Eq.~(\ref{Pu}) is invariant under the transformation
$\nu\to\tilde\nu=-\nu$ or, equivalently,
\begin{equation}
\label{ptop}
p \to \tilde{p} = {1-2p \over 2-3p} \ .
\end{equation}
The perturbation spectrum produced during a power-law inflationary
expansion with $p>1$ is indistinguishable from the spectrum produced
during a power-law collapse $\tilde{p}<1$, where $\tilde{p}$ is given
by Eq.~(\ref{ptop}). There are two fixed points where
$\tilde{p}=p$. These occur where $p=1/3$ or $p=1$, which correspond to
$\nu=0$ and $|\nu|\to\infty$ respectively.

Thus one obtains a scale invariant spectrum of perturbations not
just for de Sitter inflation in flat FRW (where
$p\to\infty$) but also for $p=2/3$, which corresponds to a collapsing
dust-dominated FRW universe.  This result is rather surprising at first
sight since the scale invariance of the de Sitter spectrum can be
understood as being due to the time-invariant nature of this solution, and a
collapsing dust universe seems to be far from static. However, there
is an important difference between the two cases. For $\nu>0$ (which
includes conventional inflation models with $p>1$) the scalar field
perturbations become frozen in on large scales as $H^2\propto\eta^{2\nu-3}$ in
Eq.~(\ref{Pdphi}). Thus ${\cal P}_{\delta\phi}$ remains constant for
any given mode $k$ as $|k\eta|\to0$. But for $\nu<0$, the
perturbations grow outside the horizon with ${\cal
P}_{\delta\phi}\propto \eta^{-4|\nu|}$ as $|k\eta|\to0$. The amplitude
of the perturbations as they cross outside the horizon ($|k\eta|=1$)
grows as $H^2\propto\eta^{2\nu-3}$, and thus the amplitude of modes already
outside the horizon grows at precisely the same rate for $\nu=-3/2$
and at any given time the spectrum is scale-invariant on
large-scales ($|k\eta|\ll1$).

If one asks what is the most general cosmological evolution which will
lead to an equivalent time-dependent mass for the perturbations and a
scale-invariant spectrum of perturbations in massless fields, one
obtains the simple solution
\begin{equation}
\label{tildeHZ}
\tilde{a}(\eta) = C_1
 \left[ \left({\eta\over\eta_1}\right)^{-1}
 + \left({\eta\over\eta_1}\right)^2 \right] \ ,
\end{equation}
which describes a non-singular metric smoothly interpolating between a
collapsing dust solution at early times ($\eta\to-\infty$) and an
exponentially expanding de Sitter solution at late times ($\eta\to0$).

One can go on to ask whether given any particular cosmological
solution $a(\eta)$ one can write down the most general evolution
$\tilde{a}(\eta)$ that would give rise to an equivalent time-dependent
mass-squared, $\mu^2$, and hence perturbation spectrum ${\cal P}_u$.
The answer turns out to be that the same spectrum of perturbations on
large scales will be produced by the two parameter family of solutions
\begin{equation}
\label{atoa}
a(\eta) \to \tilde{a}(\eta)
 = C a(\eta) \int_{\eta_*}^\eta {d\eta' \over a^2(\eta')} \ .
\end{equation}
The constant $C$ describes an arbitrary rescaling of the whole metric
which does not change the essential physics of the solutions, but the
constant of integration $\eta_*$ describes a one parameter family of
different solutions.

For example, substituting in the power-law inflationary
solutions given in Eq.~(\ref{pla}) one obtains
\begin{equation}
\label{pltildea}
\tilde{a}(\eta) = C_1 \left({\eta\over\eta_1}\right)^{1/2}
 \left[ \left({\eta\over\eta_1}\right)^{\nu}
 + \left({\eta\over\eta_1}\right)^{-\nu} \right] \ .
\end{equation}

Gravitational waves (transverse, traceless perturbations of the
metric) in Einstein gravity obey the same wave equation as a minimally
coupled massless scalar field~\cite{gw} and hence the graviton
spectrum is proportional to ${\cal P}_{\delta\phi}$. This is often
assumed to give an unambiguous record of the evolution of the
cosmological scale factor, $a(\eta)$. However an identical spectrum of
gravitational waves will be produced by the two parameter family of
solutions given in Eq.~(\ref{atoa}).

This invariance of cosmological perturbation spectra has already been
noted in the context of superstring cosmology where the perturbation
spectra of fields in the low energy string effective action may be
invariant under symmetries of the action. In the pre big bang
scenario~\cite{pbb} the graviton and dilaton fields are minimally
coupled in the conformal Einstein frame where the metric evolves as
$a\propto t^{1/3}$. $p=1/3$ is a fixed point of the transformation
given in Eq.~(\ref{ptop}) and the graviton and dilaton spectra on
large scales remain invariant under T-duality or S-duality
transformations of the background model. However the axion-type fields
are minimally coupled in the conformally related axion
frames~\cite{CEW97,CLW98}. SL(2,Z) S-duality transformations of the
power-law vacuum solutions lead to a scale factor in the axion frame
which evolves as given in Eq.~(\ref{pltildea})~\cite{CEW97}. By
constructing explicitly SL(2,R) invariant perturbation variables it
was shown that both the axion and dilaton spectra remained invariant
under arbitrary SL(2,R)
transformations~\cite{CEW97}. Equation~(\ref{atoa}) generalises this
result to arbitrary background solutions for $a(\eta)$, and to
theories which may or may not have their origin in superstring theory.

The wave equation~(\ref{ukeom}) for the perturbation $u$ may be
derived from an effective action
\begin{equation}
\label{myS}
S = {1\over2} \int d\eta \int d^3x \left\{ u'^2 - u_{,i}u_{,i} - \mu^2
u^2 \right\} \ ,
\end{equation}
with the corresponding Hamiltonian
\begin{equation}
\label{myH}
{\cal H} = {1\over2} \int d^3x \left\{ \pi_u^2 +
 u_{,i}u_{,i} + \mu^2 u^2 \right\} \ ,
\end{equation}
where the momentum canonically conjugate to $u$ is $\pi_u=u'$.
The action $S$ and Hamiltonian ${\cal H}$ both remain invariant under
the transformation given in Eq.~(\ref{atoa}) which leaves $u(\eta)$
and $\mu^2(\eta)$ invariant.
It is interesting to compare this with a different invariance which has
also recently been noted in the context of superstring
cosmology~\cite{BMUV}, and applied to generalised cosmological
perturbations~\cite{BGV}. This is an invariance of the effective
action
\begin{equation}
\label{theirS}
\hat{S} = {1\over2} \int d\eta \int d^3x \, a^2 \left\{ \delta\phi'^2 -
 \delta\phi_{,i}\delta\phi_{,i} \right\} \ ,
\end{equation}
and corresponding Hamiltonian
\begin{equation}
\label{theirH}
\hat{{\cal H}} = {1\over2} \int d^3x \left\{ a^{-2}
 \pi_{\delta\phi}^2 + a^2 \delta\phi_{,i}\delta\phi_{,i} \right\} \ ,
\end{equation}
written in terms of the field perturbation $\delta\phi$ and its
conjugate momentum $\pi_{\delta\phi}=a^2\delta\phi'$.
The Lagrangian in Eq.~(\ref{theirS}) differs from that in
Eq.~(\ref{myS}) by a total derivative
\begin{equation}
\hat{S} = S - {1\over2} \int d\eta \int d^3x \, {d \over d\eta} \left(
{a'\over a} u^2 \right) \ ,
\end{equation}
which does not affect the equation of motion, Eq.~(\ref{eom}), but does
change the Hamiltonian
\begin{equation}
\hat{{\cal H}} = {\cal H} + \int d^3x \left\{ {a''\over a}u^2 - {1\over2}
\left( {a'\over a} u^2 \right)^{\prime} \right\} \ .
\end{equation}
There is a duality invariance of the action $\hat{S}$ and Hamiltonian
$\hat{{\cal H}}$ under which the ``pump field''
$a^2\to\tilde{a}^2=a^{-2}$ is inverted and the field perturbation
$\delta\phi$ is exchanged with its canonical momentum
$\pi_{\delta\phi}$~\cite{BGV}.
The Hamiltonian $\hat{{\cal H}}$ does not remain invariant under the
transformation in Eq.~(\ref{atoa}), but neither does the Hamiltonian
${\cal H}$ remain invariant under the duality transformation in
Ref.\cite{BGV}. The effective action is only defined up to boundary
terms and due to the explicit time-dependence of $a(\eta)$, the
Hamiltonian is not uniquely defined. Both transformations, however, represent
symmetries of the equation of motion.

The most cosmologically significant perturbation spectrum
produced during an inflationary era in the early universe is likely to
be the primordial spectrum of adiabatic density perturbations on large
scales induced by the perturbations in the scalar field which drives
inflation. To study the evolution of this field requires us to include
the self-interaction potential of the field and the back-reaction of
metric fluctuations. Fortunately Mukhanov~\cite{Mukhanov} has shown
that the wave equation for the gauge invariant field perturbation
\begin{equation}
u = a \left[ \delta\phi + \dot\phi {\psi \over H} \right] \ ,
\end{equation}
where $\psi$ is the gauge-dependent curvature perturbation~\cite{MFB},
can still be written in the form given in Eq.~(\ref{ukeom}) but with a
time-dependent mass-squared
\begin{equation}
\label{muz}
\mu^2 = -{z'' \over z} \ ,
\end{equation}
where $z=a\dot\phi/H$.
Quite generally we can write
\begin{equation}
\label{zgamma}
z = a \sqrt{ {3\gamma \over 8\pi G} } \ ,
\end{equation}
where the effective barotropic index
$\gamma\equiv\dot\phi^2/(V+\dot\phi^2/2)$. In the special case of
power-law inflation driven by a scalar field with exponential
potential, $\dot\phi\propto H$ and hence $\gamma$ is a constant and we
have $z\propto a$.

Starting from any known solution $z(\eta)$ we obtain the identical
spectrum of perturbations ${\cal P}_u$ from the two parameter family
of solutions
\begin{equation}
\label{ztoz}
z(\eta) \to \tilde{z}(\eta)
 = C z(\eta) \int_{\eta_*}^\eta {d\eta' \over z^2(\eta')} \ .
\end{equation}
which leaves $\mu^2$ given in Eq.~(\ref{muz}) invariant.

The gauge-invariant curvature perturbation $\zeta$~\cite{BST,MFB,LL93}
is related to the field perturbation $u$ by
\begin{equation}
\zeta = \psi + {H\over\dot\phi} \delta\phi = {u \over z} \ .
\end{equation}
This is usually evaluated in terms of the quantities at horizon
crossing. This is because $\zeta$ becomes constant on super-horizon
scales for adiabatic perturbations. In this case the $z$ acquires an
implicit scale dependence due to the different times at which
different scales are evaluated. However one can also evaluate $\zeta$
at a fixed time, such as the end of inflation, in which case the scale
dependence of $\zeta$ is due solely to the scale dependence of $u$,
and $z$ contributes a scale independent factor.
Thus under the transformation given by Eq.~(\ref{ztoz}) the curvature
perturbation is rescaled by an overall factor $z/\tilde{z}$, but the
spectral index
\begin{equation}
n \equiv 1 + {d\ln {\cal P}_\zeta \over d\ln k} \ ,
\end{equation}
remains invariant.

For instance, it is well-known that the extreme slow-roll limit of
inflation corresponding de Sitter expansion driven by a massless
scalar field, where $z\propto \eta^{-1}$, leads to a scale-invariant
Harrison-Zel'dovich ($n=1$) spectrum of curvature
perturbations. However substituting this familiar form for $z(\eta)$
into Eq.~(\ref{ztoz}) yields the most general evolution which gives a
scale-invariant spectrum as
\begin{equation}
\label{altHZ}
\tilde{z}(\eta) = C_1
 \left[ \left({\eta\over\eta_1}\right)^{-1}
 + \left({\eta\over\eta_1}\right)^2 \right] \ .
\end{equation}
This shows that it is in fact possible to produce a scale-invariant
spectrum of curvature perturbations from inflation that is far from
the usual slow-roll limit.

Unfortunately it is not possible to uniquely determine the form of the
self-interaction potential $V(\phi)$ for a given $z(\eta)$, such as
that given in Eq.~(\ref{altHZ}). For example, both power-law
inflation~\cite{PLI} driven by an exponential potential, and natural
inflation~\cite{natural} where the potential energy remains
effectively constant, can give rise to a power-law spectrum of
curvature perturbations with $n=$constant~\cite{SL}.  However it is
possible to test the consistency of the slow-roll approximation for a
given $z(\eta)$.  The slow-roll approximation requires that the
effective barotropic index, $\gamma$ in Eq.~(\ref{zgamma}), is small
and slowly varying, so that to zeroth order in the slow-roll
parameters~\cite{recon}, the evolution of $z$ is determined by the
growth of the scale factor $a\sim\eta^{-1}$. This implies that
$z''z/z^{\prime2}\approx2$.  For the general form of
$\tilde{z}(\eta)$ which yields a scale-invariant spectrum of curvature
perturbations, given in Eq.~(\ref{altHZ}), this slow-roll condition is
badly broken at early times for $|\eta/\eta_1|\gg1$.

Even in the slow-roll limit, the spectrum of curvature perturbations
is sufficient only to determine the inflaton potential up to a one
parameter class of solutions~\cite{recon}. The amplitude of the
gravitational wave perturbations is then required to fix the actual
amplitude of the inflaton potential. In this paper I have demonstrated
that if one allows behaviour which may be far from the slow-roll limit
there is a degeneracy even in the spectrum of gravitational wave
perturbations.  The general solution which yields an almost
scale-invariant spectrum of gravitational waves interpolates between
an initially collapsing universe and a quasi-de Sitter
expansion. However, the asymptotic behaviour at late times reproduces
the usual slow-roll result, so in practice this need not be a serious
limitation for reconstructing the evolution in the context of
conventional inflation models~\cite{recon}.

On the other hand the transformation presented here suggests that it
might be possible to use slow-roll techniques to analyse perturbations
in models far from the usual slow-roll limit if they can be related to
slow-roll models.  An example of this is provided by solutions to the
low energy string effective action where perturbation spectra in
general axion-dilaton cosmologies can be related to much simpler
dilaton-vacuum solutions by a duality transformation~\cite{CEW97}.

\acknowledgements

DW would like to thank John Barrow, Ed Copeland, Andrew Liddle, Jim
Lidsey and Karim Malik for helpful comments.

\vspace*{-12pt}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
\citet{Dolinsky} showed a Donsker-type result for $G$-Brownian motion
by introducing a notion of volatility uncertainty in discrete time and defined a discrete version of \textit{Peng's} $G$-expectation.
In the continuous-time limit, the resulting sublinear expectation converges weakly to $G$-expectation. In their discretization, \citet{Dolinsky} allow for martingale laws whose support is the whole set of reals in a $d$-dimensional setting. In other words, they only discretize the time line, but not the state space of the canonical process. Now for certain applications, for example, a hyperfinite construction of $G$-expectation in the sense of Robinsonian nonstandard analysis, a discretization of the state space would be necessary.
Thus, we develop a modification of the construction by \citet{Dolinsky} which even ensures that the sublinear expectation operator for the discrete-time canonical process corresponding to this discretization of the state space
(whence the martingale laws are supported by a finite lattice only) converges to the $G$-expectation. Further, we prove a lifting theorem, in the sense of Robinsonian nonstandard analysis, for the $G$-expectation. Herein, we use the discretization result for the $G$-expectation.

Nonstandard analysis makes consistent use of infinitesimals in mathematical analysis based on techniques from mathematical logic. This approach is very promising because it also allows, for instance, to study continuous-time stochastic
processes as formally finite objects. Many authors have applied nonstandard analysis to problems in measure theory, probability theory and mathematical economics (see for example, \citet{Raimondo} and the references therein or the
contribution in \citet{Imme}), especially after \citet{Loeb} converted nonstandard measures (i.e. the images of standard measures under the nonstandard embedding ${}^{*}$) into real-valued,
countably additive measures, by means of the standard part operator and \textit{Caratheodory}'s extension theorem.
One of the main ideas behind these applications is the extension of the notion of a finite set known as {\em hyperfinite set} or more causally,
a formally finite set. Very roughly speaking, hyperfinite sets are sets that can be formally enumerated with both standard and nonstandard natural numbers up to a (standard or nonstandard, i.e. unlimited) natural number.

\citet{Anderson, Keisler, Toml, Edwin}, a few to mention, used Loeb's \cite{Loeb} approach to develop basic nonstandard stochastic analysis and in particular, the nonstandard It{\^{o}} calculus. \citet{Loeb} also presents the construction of a Poisson processes using nonstandard analysis.
\citet{Anderson} showed that Brownian motion can be constructed from a hyperfinite number of coin tosses, and provides a detailed proof using a special case of Donsker's theorem. \citet{Anderson} also gave a nonstandard construction of stochastic
integration with respect to his construction of Brownian motion. \citet{Keisler} uses Anderson's \cite{Anderson} result to obtain some results on stochastic differential equations. \citet{Toml} gave the hyperfinite construction ({\em lifting}) of $L^2$ standard martingales.
Using nonstandard stochastic analysis, \citet{Perkins81} proved a global characterization of (standard) Brownian local time.
In this paper, we do not work on the Loeb space because the $G$-expectation and its corresponding $G$-Brownian motion are not based on a classical probability measure, but on a set of martingale laws.

The aim of this paper is to give two approximation results on $G$-expectation. First, to refine the discretization of $G$-expectation by \citet{Dolinsky}, in order to obtain a discretization of  the sublinear expectation where the martingale laws are defined on a finite lattice rather than the whole set of reals. Second, to give an alternative, combinatorially inspired construction of the $G$-expectation based on the discretization result. We hope that this result may eventually become useful for applications in financial economics (especially existence of equilibrium on continuous-time financial markets with volatility uncertainty) and provides additional intuition for {\em Peng}'s $G$-stochastic calculus.
We begin the nonstandard treatment of the $G$-expectation by defining a notion of $S$-continuity, a standard part operator, and proving a corresponding lifting (and pushing down) theorem. Thereby, we show that our hyperfinite construction is the appropriate nonstandard analogue of the $G$-expectation.

The rest of this paper is divided into two parts: in the first part, Section \ref{weakapprox}, we define Peng's $G$-expectation and introduce a discrete-time analogue of a $G$-expectation in the spirit of \citet{Dolinsky}. Unlike in \citet{Dolinsky}, we require the discretization of the martingale laws to be defined on a finite lattice rather than the whole set of reals. In the continuous-time limit, the resulting sublinear expectation converges weakly to the continuous-time $G$-expectation.
In the second part, Section \ref{hyperfinitecons},   we develop the basic theory for hyperfinite $G$-expectations and prove an existence theorem for liftings of (continuous-time) $G$-expectation. We extend the discrete time analogue of the $G$-expectation in Section \ref{weakapprox} to a hyperfinite time analogue. Then, we use the characterization of convergence in nonstandard analysis to prove that the hyperfinite discrete-time analogue of the $G$-expectation is infinitely close in the sense of nonstandard topology to the continuous-time $G$-expectation.

\section{Weak approximation of $G$-expectation with discrete state space}
\label{weakapprox}

\citet{Peng} introduced a sublinear expectation on a well-defined space $\mathbb{L}^1_{G}$, the completion of $\text{Lip}_{b.cyl}(\Omega)$ (bounded and Lipschitz cylinder function) under the norm $\| \cdot \|_{\mathbb{L}^1_{G}}$,
under which the increments of the canonical process $(B_t)_{t>0}$ are zero-mean, independent and stationary and can be proved to be $(G)$-normally distributed. This type of process is called {\em G-Brownian motion}
and the corresponding sublinear expectation is called {\em G-expectation}.

The $G$-expectation $\xi \mapsto \mathcal{E}^{G}(\xi)$ is a sublinear operator defined on a class of random variables on $\Omega$. The symbol $G$ refers to a given function
\begin{equation}
\label{gfunc}
G(\gamma):= \frac{1}{2} \sup_{c \in \mathbf{D}} c \gamma :\mathbb{R} \rightarrow \mathbb{R}
\end{equation}
where $\mathbf{D}=[r_{\mathbf{D}},R_{\mathbf{D}}]$ is a nonempty, compact and convex set, and $0\leq r_{\mathbf{D}} \leq R_{\mathbf{D}} < \infty$ are fixed numbers. The construction of the $G$-expectation is as follows. Let $\xi = f(B_T)$,
where $B_T$ is the $G$-Brownian motion and $f$ a sufficiently regular function. Then $\mathcal{E}^{G}(\xi)$ is defined to be the initial value $u(0,0)$ of the solution of the nonlinear backward heat
equation,
$$-\partial_t u - G(\partial^2_{xx} u) = 0,$$
with terminal condition $u(\cdot, T)=f$, \citet{Pardoux}. The mapping $\mathcal{E}^{G}$ can be extended to random variables of the form ${\xi = f(B_{t_1}, \cdots, B_{t_n} )}$ by a stepwise evaluation of the PDE and then to the completion $\mathbb{L}^1_{G}$ of the space of all such random variables (cf. \citet{Dolinsky}).
\citet{Denis} showed that $\mathbb{L}^1_{G}$ is the completion of $\mathcal{C}_b(\Omega)$ and $\text{Lip}_{b.cyl}(\Omega)$ under the norm ${\| \cdot \| _{\mathbb{L}^1_{G}}}$, and that $\mathbb{L}^1_{G}$ is the space of the so-called quasi-continuous function and contains all bounded continuous functions on the canonical space $\Omega$, but not all bounded measurable functions are included. \citet{Ruan} introduced the invariance principle of $G$-Brownian motion using the theory of sublinear expectation.
There also exists an equivalent alternative representation of the $G$-expectation known as the {\em dual view on $G$-expectation via volatility uncertainty}, see \citet{Denis}:
\begin{equation}
\label{superhedge}
\mathcal{E}^{G}(\xi) = \sup _{P \in \mathcal{P}^G} \mathbb{E}^{P} [\xi], \quad \xi= f(B_T),
\end{equation}
where $\mathcal{P}^G$ is defined as the set of probability measures on $\Omega$ such that, for any $P \in \mathcal{P}^G$, $B$ is a martingale with
the volatility $d\left\langle B \right\rangle _t / dt \in \mathbf{D}$ $P \otimes dt$ a.e.

\subsection{Continuous-time construction of sublinear expectation}
\label{G-framework}
Let $\Omega = \{ \omega \in \mathcal{C}([0,T]; \mathbb{R}) : \omega_0=0 \}$ be the canonical space endowed with the uniform norm ${\|\omega\| _{\infty} = \sup _{0 \leq t \leq T} |\omega _t| },$ where $|\cdot|$
denotes the absolute value
on $\mathbb{R}$. Let $B$ be the canonical process $B_t(\omega) = \omega_t,$ and $\mathcal{F}_t = \sigma (B_s, 0 \leq s \leq t)$
the filtration generated by $B$.
A probability measure $P$ on $\Omega$ is a martingale law provided $B$ is a $P$-martingale and $B_0 = 0$ $P$ a.s. Then, $\mathcal{P}_{\mathbf{D}}$ is the set of martingale laws on $\Omega$ and
the volatility takes values in $\mathbf{D}$, $P \otimes dt$ a.e;
\begin{equation*}
\mathcal{P}_{\mathbf{D}} = \left\{ P \text{ martingale law on } \Omega \text{: } d\left\langle B \right\rangle _t / dt \in \mathbf{D} \text{, } P \otimes dt \text{ a.e.} \right\}.
\end{equation*}

\subsection{Discrete-time construction of sublinear expectation}We denote
\begin{equation*}
\mathcal{L}_n = \left\{ \frac{j}{n \sqrt{n}}, \quad -n^2 \sqrt{R_{\mathbf{D}}} \leq j \leq  n^2 \sqrt{R_{\mathbf{D}}}, \quad \text{for } j\in \mathbb{Z}\right\},
\end{equation*}
and $\mathcal{L}^{n+1}_{n}= \mathcal{L}_n \times \dots \times \mathcal{L}_n  (n+1 \text{  times})$, for $n\in \mathbb{N}$.
Let $X^n = {(X^n_k)}_{k=0}^{n}$ be the canonical process $X^n_k(x)= x_k$ defined on $\mathcal{L}^{n+1}_{n}$ and ${(\mathcal{F}_k^n)}_{k=0}^{n} = \sigma (X_l^n, l =0, \dots, k)$ be the filtration generated by $X^n$. We note that ${R_{\mathbf{D}} = \sup_{\alpha \in \mathbf{D}}|\alpha|}$.
$$\mathbf{D}^{\prime}_n = \mathbf{D} \cap \left( \frac{1}{n} \mathbb{N} \right)^2$$
is a nonempty bounded set of volatilities. A probability measure $P$ on $\mathcal{L}_{n}^{n+1}$ is a martingale law provided $X^n$ is a $P$-martingale and $X^n_0=0$ $P$ a.s. The increment $\Delta X^n_k = X^n_k - X^n_{k-1}$.
Let $\mathcal{P}_{\mathbf{D}}^{n}$ be the set of martingale laws of $X^n$ on $\mathbb{R}^{n+1} $, i.e.,
\begin{equation*}
\mathcal{P}_{\mathbf{D}}^{n} = \left\{ P \text{ martingale law on } \mathbb{R}^{n+1}  \text{: } r_{\mathbf{D}} \leq |\Delta X^n_k|^2 \leq R_{\mathbf{D}} \text{, }P \text{ a.s.} \right\},
\end{equation*}
such that for all $n$, $ \mathcal{L}_{n}^{n+1} \subseteq \mathbb{R}^{n+1} $.
\subparagraph*{ }In order to establish a relation between the continuous-time and discrete-time settings, we obtained a continuous-time process $\widehat{x} _t \in \Omega$ from any discrete path $x \in \mathcal{L}^{n+1}_{n}$ by linear interpolation. i.e.,
$$\widehat{x}_t := (\lfloor nt/T \rfloor +1 -nt/T)x_{ \lfloor nt/T \rfloor } + (nt/T - \lfloor nt/T \rfloor)x_{\lfloor nt/T \rfloor +1}$$
where $\widehat{   }: \mathcal{L}^{n+1}_{n} \rightarrow \Omega $ is the linear interpolation operator, $x = (x_0, \dots, x_n) \mapsto \widehat{x} = \{(\widehat{x})_{0\leq t \leq T}\} ,$ and $ \lfloor y \rfloor $ denotes the greatest integer less than or equal to $y.$ If $X^n$ is the canonical process on $\mathcal{L}^{n+1}_{n}$ and $\xi$
is a random variable on $\Omega,$ then $\xi (\widehat{X}^n)$ defines a random variable on $\mathcal{L}^{n+1}_{n}.$

\subsection{Strong formulation of volatility uncertainty}
We consider martingale laws generated by stochastic integrals with respect to a fixed Brownian motion as in \citet{Dolinsky,Marcel} and a fixed random walk as in \citet{Dolinsky}.\\
Continuous-time construction; let $\mathcal{Q}_{\mathbf{D}} $ be the set of martingale laws:
\begin{equation*}
\mathcal{Q}_{\mathbf{D}} = \left\{ P _{0} \circ (M)^{-1}; \text{ } M = \int f (t, B) dB_t, \text{ and }  f \in \mathcal{C} ([0,T] \times \Omega ; \sqrt{\mathbf{D}}) \text{ is adapted} \right\}.
\end{equation*}
$B$ is the canonical process under the Wiener measure $P_0 $.\\
Discrete-time construction; we fix $n \in \mathbb{N}$, $\Omega _n = \lbrace \omega = (\omega _1, \dots, \omega _n) : \omega _i \in \lbrace \pm 1 \rbrace, \quad i =1, \dots,n \rbrace$ equipped with the power set and let
$$P_n = \underbrace{ \frac{\delta_{-1}+ \delta_{+1}}{2} \otimes \cdots \otimes  \frac{\delta_{-1}+ \delta_{+1}}{2} }_\text{n times}$$
be the product probability associated with the uniform distribution where $\delta_{x}(A)$ is a Dirac
measure for any $A \subseteq \mathbb{R}$ and a given $x \in A$.
Let $\xi _1, \dots, \xi_n$ be an i.i.d sequence of $\lbrace \pm 1 \rbrace$-valued random variables. The components of $\xi _k$ are orthonormal in $L^2(P_n)$ and the associated scaled random walk is $$ \mathbb{X} = \frac{1}{\sqrt{n}} \sum _{l=1}^k \xi _l. $$
We denote by $\mathcal{Q}_{\mathbf{D}^{\prime}_n}^{n}$ the set of martingale laws of the form:
\begin{align}
\label{disc-meas}
\mathcal{Q}_{\mathbf{D}^{\prime}_n}^{n} = \left\{  P_n  \circ (M^{f, \mathbb{X}})^{-1}; \text{  } f: \lbrace 0, \dots, n \rbrace \times \mathcal{L}^{n+1}_{n} \rightarrow \sqrt{\mathbf{D}^{\prime}_n} \text{ is } \mathcal{F}^n \text{-adapted.}
\right\}
\end{align}
where $ M^{f, \mathbb{X}} = \left( \sum _{l=1}^{k} f (l-1,\mathbb{X})\Delta \mathbb{X}_ l  \right)_{k=0}^{n}.$

\subsection{Results and proofs}
\label{G-result}
Theorem \ref{max-result}  states that a sublinear expectation with discrete-time volatility uncertainty on our finite lattice converges to the $G$-expectation.

\begin{lem}
	\label{equivmeas}

	$\mathcal{Q}_{\mathbf{D}}^{n} = \left\{ P_n  \circ \left( M^{f,\mathbb{X}} \right) ^{-1}; \text{ }f: \lbrace 0, \dots, n \rbrace \times \mathbb{R}^{n+1} \rightarrow \sqrt{\mathbf{D}} \text{ is adapted}\right\}.
	$ Then $\mathcal{Q}_{\mathbf{D}}^{n} \subseteq \mathcal{P}_{\mathbf{D}}^{n}$.
\end{lem}

\begin{pro}
	\label{mthm}

	Let $\xi: \Omega \rightarrow \mathbb{R}$ be a continuous function satisfying $|\xi(\omega)| \leq a(1+\parallel \omega\parallel_{\infty})^b$ for some constants $a,b >0.$ Then,
	\begin{itemize}
		\item[$(i)$]
		\begin{equation}
		\label{main-thm}
		\lim _{n \rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)]  = \sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi].
		\end{equation}
		\item[$(ii)$]
		\begin{equation}
		\label{max-eq}
		\sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] =  \max_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)].
		\end{equation}
	\end{itemize}
\end{pro}

To prove \eqref{main-thm}, we prove two separate inequalities together with a density argument.
The left-hand side of \eqref{max-eq} can be written as
\begin{equation*}
\sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] =  \sup_{f \in \mathcal{A} } \mathbb{E}^{P_n \circ (M^{f, \mathbb{X}})^{-1}} [\xi (\widehat{X}^n)],
\end{equation*}
where $\mathcal{A} =  \left\lbrace f: \lbrace 0, \dots, n \rbrace \times \mathcal{L}^{n+1}_{n} \rightarrow \sqrt{\mathbf{D}^{\prime}_n/n}\text{ is } \mathcal{F}^n \text{-adapted.}\right\rbrace$. We prove that $\mathcal{A}$ is a compact subset of a finite-dimensional
vector space, and that $f\mapsto \mathbb{E}^{P_n \circ (M^{f, \mathbb{X}})^{-1}} [\xi (\widehat{X}^n)]$ is continuous. Before then, we introduce a smaller space $\mathbb{L}_{*}^1$ that is defined as the completion of $\mathcal{C}_b (\Omega; \mathbb{R})$ under the norm
{(cf. \citet{Dolinsky})}
\begin{equation*}
\parallel \xi \parallel _{*} := \sup _{Q \in \mathcal{Q}} \mathbb{E} ^{Q} |\xi| , \quad \mathcal{Q}:=  \mathcal{P}_{\mathbf{D}} \cup \lbrace P \circ (\widehat{X}^n)^{-1};  P \in \mathcal{P}_{\mathbf{D}/n}^{n}, { }  n \in \mathbb{N}. \rbrace .
\end{equation*}
This is because Proposition \ref{mthm} will not hold if $\xi$ just belong to $\mathbb{L}^1_{G}$, which is the completion of $\mathcal{C}_b (\Omega; \mathbb{R})$ under the norm
\begin{equation}
\label{dnorm}
\parallel \xi \parallel _{\mathbb{L}^1_{G}} := \sup _{P \in \mathcal{P}_{\mathbf{D}}} \mathbb{E} ^{P} [|\xi|].
\end{equation}

\begin{proof}[Proof of Proposition \ref{mthm}]
	\textit{First inequality \textnormal{(for $\leq$ in \eqref{main-thm})}:}
	\begin{equation}
	\label{first-inequality}
	\limsup _{n \rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)]  \leq  \sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi].
	\end{equation}
	For all $n$, $\sqrt{\mathbf{D}^{\prime}_n/n} \subseteq \sqrt{\mathbf{D}/n}$ and $\mathcal{Q}_{\mathbf{D}^{\prime}_n}^{n} \subseteq \mathcal{Q}_{\mathbf{D}}^{n}$.
	It is shown in \citet{Dolinsky} that
	\begin{equation*}
	\limsup_{n\rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{P}_{\mathbf{D}/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] \leq \sup_{P \in \mathcal{P}_{\mathbf{D}}} \mathbb{E}^{P} [\xi].
	\end{equation*}
	Since $\mathcal{Q}_{\mathbf{D}} \subseteq \mathcal{P}_{\mathbf{D}}$ (see \citet[Remark~$3.6$]{Dolinsky})
	and $\mathcal{Q}^{n}_{\mathbf{D}} \subseteq \mathcal{P}^{n}_{\mathbf{D}}$ (see Lemma \ref{equivmeas}), \eqref{first-inequality} follows.

	\textit{Second inequality \textnormal{(for $\geq$ in \eqref{main-thm})}:} It remains to show that
	$$\liminf _{n \rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)]  \geq  \sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi].$$
	For arbitrary $P \in \mathcal{Q}_{\mathbf{D}}$, we construct a sequence ${(P^n)}_n$ such that for all  $n$,
	\begin{equation}
	\label{new-sq}
	P^n \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n},
	\end{equation}
	and
	\begin{equation}
	\label{second}
	\mathbb{E}^{P}[\xi] \leq \liminf _{n\rightarrow \infty} \text{  } \mathbb{E}^{P^n}[\xi (\widehat{X}^n)].
	\end{equation}
	For fixed $n$, we want to construct martingales $M^n$ whose laws are in $\mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}$ and the laws of their interpolations tend to $P.$
	Thus, we introduce a scaled random walk with the piecewise constant c{\`a}dl{\`a}g property,

	\begin{equation}
	\label{pr}
	W^n_t := \frac{1}{\sqrt{n}} \sum _{l=1}^{ \lfloor nt/T \rfloor } \xi _l = \frac{1}{\sqrt{n}} Z^n_{\lfloor nt/T \rfloor}, \quad 0\leq t \leq T,
	\end{equation}
	and we denote the continuous version of \eqref{pr} obtained by linear interpolation by
	\begin{equation}
	\label{pir}
	\widehat{W}^n_t := \frac{1}{\sqrt{n}} \widehat{Z}^n_{\lfloor nt/T \rfloor}, \quad 0\leq t \leq T.
	\end{equation}
	By the central limit theorem; $(W^n, \widehat{W}^n) \Rightarrow (W,W) $ as $n \rightarrow \infty$ on $D([0,T];\mathbb{R}^2)$ ($\Rightarrow$ implies convergence in distribution). i.e., the law $(P_n) $ converges to the law $P_0$ on the Skorohod space $D([0,T];\mathbb{R}^2)$
	\citet[Theorem~$27.1$]{Patrick}. Let $g \in \mathcal{C}([0,T]\times \Omega, { } \sqrt{\mathbf{D}})$ such that
	$$P = P_0 \circ \left( \underbrace{ \int g(t, W) dW_t}_\text{$M$} \right)^{-1}.$$
	Since $g$ is continuous and $\widehat{W}^n_t$ is the interpolated version of \eqref{pr},
	$$\left( W^n, \left( g\left( \lfloor nt/T \rfloor T/n, \widehat{W}^n_t \right) \right)_{t \in [0,T]} \right) \Rightarrow \left(W, (g(t, W_t))_{t \in [0,T]} \right) \text{ as } n \rightarrow \infty \text{ on } D([0,T];\mathbb{R}^2).$$
	We introduce martingales with discrete-time integrals,
	\begin{equation}
	\label{dis-mart}
	M^{n}_k := \sum _{l=1}^{k} g \left( (l-1)T/n, \widehat{W}^n \right) \widehat{W}^n_{lT/n} - \widehat{W}^n_{(l-1)T/n}.
	\end{equation}
	In order to construct $M^n$ which is ``close'' to $M$ and also is such that ${P_n \circ {(M^n)}^{-1}  \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}}$. We choose ${\widetilde{h}_n: \{0, \cdots, n\} \times \Omega \rightarrow {\sqrt{{\mathbf{D}^\prime_n}/n}}} $ such that
	$$d_{J_1}\left( \left( \widetilde{h}_n (\lfloor nt/T \rfloor T/n, \widehat{W}^{n}_{t}) \right)_{t\in[0,T]} ,   \left( g (\lfloor nt/T \rfloor T/n, \widehat{W}^{n}_{t}) \right)_{t\in[0,T]}                    \right) $$
	is minimal (this is possible because there are only finitely many choices for $\left( \widetilde{h}_n (\lfloor nt/T \rfloor T/n, \widehat{W}^{n}_{t}) \right)_{t\in[0,T]} )$
	and $d_{J_1}$ is the Kolmogorov metric for the Skorohod $J_1$  topology.
	From \citet[Theorem~$4.3$ and Definition~$4.1$]{Billingsley}, it follows that
	$$ \left( W^n, \left( \widetilde{h}_n \left(\lfloor nt/T \rfloor T/n, \widehat{W}^{n}_{t} \right) \right)_{t\in[0,T]} \right) \Rightarrow \left(W, g(t, W_t)_{t \in [0,T]} \right) \text{ on }D([0,T];\mathbb{R}^2).$$
	We then define $g_n: \lbrace 0, \dots, n \rbrace \times \mathcal{L}^{n+1}_{n} \rightarrow {\sqrt{\mathbf{D}^{\prime}_n/n}}$ by
	$g_n:( \ell, \vec{\mathbb{X}}) \mapsto \widetilde{h}_n ( \ell , \widehat{\vec{X}}).$
	Let $M^n$ be defined by
	$$M^{n}_k = \sum _{l=1}^{k} g_n \left(l-1, \frac{1}{\sqrt{n}} Z^n \right) \frac{1}{\sqrt{n}} \Delta Z^n_l, \quad \forall k \in \{0,\cdots, n\}.$$
	By stability of stochastic integral (see \citet[Theorem~$4.3$ and Definition~$4.1$]{Duffie}),
	$$\left( M^n_{\lfloor nt/T \rfloor} \right) _{t \in [0,T]} \Rightarrow M \quad \text{as } n \rightarrow \infty \text{ on }  D([0,T];\mathbb{R})$$
	because
	$$M^n_{\lfloor nt/T \rfloor} = \sum_{l=1}^{\lfloor nt/T \rfloor} \widetilde{h}_n \left((l-1)T/n, \left(\widehat{W}_{kT/n} \right)_{k=0}^{n} \right) \Delta \widehat{W}_{lT/n}. $$
	In addition, as $n$ goes to $\infty$, the increments of $M^n$ uniformly tend to $0$. Thus, $\widehat{M}^n \Rightarrow M$ on $\Omega.$ Since $\xi$ is bounded and continuous,
	\begin{equation}
	\label{sk-conv}
	\lim_{n \rightarrow \infty} \mathbb{E}^{P_n \circ {(M^n)}^{-1}}[\xi(\widehat{X}^n)] = \mathbb{E}^{P_0 \circ {M}^{-1}}[\xi].
	\end{equation}
	Therefore, \eqref{new-sq} is satisfied for $P^n = P_n \circ {(M^n)}^{-1}  \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}.$ Taking the $\liminf$ as $n$ tends to $\infty$ and the supremum over $P \in \mathcal{Q}_{\mathbf{D}},$
	\eqref{sk-conv} becomes
	\begin{equation}
	\label{second-inequality}
	\sup _{ P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P}[\xi] \leq \liminf _{n \rightarrow \infty} \sup _{ \mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}}[\xi (\widehat{X}^n)].
	\end{equation}
	Combining \eqref{first-inequality} and \eqref{second-inequality},
	\begin{align*}
	{\sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi]} & \geq {\limsup _{n \rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)]}\geq {\liminf _{n \rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] } \geq {\sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi]}.
	\end{align*}
	Therefore,
	\begin{equation}
	\label{final-equation}
	\sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi] = \lim _{n\rightarrow \infty} \sup_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] .
	\end{equation}
	\textit{Density argument}: \eqref{main-thm} is established for all $\xi \in \mathcal{C}_b (\Omega, \mathbb{R})$.
	Since $\mathcal{Q}_{\mathbf{D}} \subseteq \mathcal{P}_{\mathbf{D}}$ (see \citet[Remark~$3.6$]{Dolinsky}) and $\mathcal{Q}^{n}_{\mathbf{D}} \subseteq \mathcal{P}^{n}_{\mathbf{D}}$ (see Lemma \ref{equivmeas}),
	$\mathcal{Q}_{\mathbf{D}^{\prime}_n}^{n}  \subseteq \mathcal{Q} $ and $\mathcal{Q}_{\mathbf{D}} \subseteq \mathcal{Q}$. Thus, \eqref{main-thm} holds for all $\xi \in \mathbb{L}^1_{*}$, and hence, holds for all $\xi$ that satisfy condition of Proposition \ref{mthm}.

	\textit{First part of \ref{max-eq}}:
	$\mathcal{A}$ is closed and obviously bounded with respect to the norm $\| \cdot\|_{\infty}$ as $\mathbf{D}^{\prime}_n$ is bounded. By Heine-Borel theorem, $\mathcal{A}$ is a compact subset of a $N(n,n)$-dimensional vector space\footnote{The cardinality of $\mathcal{L}_n$, $\# \mathcal{L}_n = 2n+1$, $ \# \mathcal{L}_n^{n+1} = (2n+1)^{n+1}$, and $\# (\{0, \dots, n\} \times \mathcal{L}_n^{n+1} ) = (n+1)(2n+1)^{n+1} = N(n,n).$
	} equipped with the norm ${\|\cdot\|_{\infty}}$.

	\textit{Second part of \ref{max-eq}}:
	Here, we show that $F: f \mapsto \mathbb{E}^{P_n  \circ (M^{f, \mathbb{X}})^{-1}} [\xi (\widehat{X}^n)]$ is continuous.
	From Proposition \ref{mthm} we know that $\xi$ is continuous, $\widehat{X}^n$ is the interpolated canonical process, i.e., $\widehat{X}: \mathcal{L}^{n+1}_{n} \rightarrow \Omega,$ thus $\widehat{X}^n$ is continuous and $P_n$ takes it values from the set of real numbers.
	For $F:f \mapsto \mathbb{E}^{P_n \circ (M^{f, \mathbb{X}})^{-1}} [\xi (\widehat{X}^n)]$ to be continuous, ${\psi: f \mapsto M^{f, \mathbb{X}} }$ has to be continuous.
	Since $\mathcal{A}$ is a compact subset of a $N(n,n)$-dimensional vector space for fixed $n\in \mathbb{N}$ and ${M^{f, \mathbb{X}}: \Omega_n \rightarrow \mathcal{L}^{n+1}_{n}}$, for all $f,g \in \mathcal{A}$,
	$$|M^{f, \mathbb{X}}-M^{g, \mathbb{X}}| = | \|f \| _{\infty}-  \|g \| _{\infty} | \leq \| f-g \|_{\infty}.$$
	Thus, $\psi$ is continuous with respect to the norm $\| \cdot \|_{\infty}$. Hence $F$ is continuous with respect to any norm on $\mathbb{R}^{N(n,n)}$.

\end{proof}

\begin{thm}
	\label{max-result}

	Let $\xi: \Omega \rightarrow \mathbb{R}$ be a continuous function satisfying $|\xi(\omega)| \leq a(1+\| \omega\|_{\infty})^b$ for some constants $a,b >0.$ Then,
	\begin{equation}
	\label{max-thm}
	\sup_{P \in \mathcal{Q}_{\mathbf{D}}} \mathbb{E}^{P} [\xi] = \lim _{n\rightarrow \infty} \max_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n/n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] .
	\end{equation}
\end{thm}
\begin{proof}
	The proof follows directly from Proposition \ref{mthm}.
\end{proof}

\section{Nonstandard construction of $G$-expectation}

\label{hyperfinitecons}

\subsection{Hyperfinite-time setting}
Here we present the nonstandard version of the discrete-time setting of the sublinear expectation and the strong formulation of volatility uncertainty on the hyperfinite timeline.
\begin{dfn}

	${}^{*} \Omega$ is the ${}^{*}$-image of $\Omega$ endowed with the ${}^{*}$-extension of the maximum norm $ {}^{*} \|\cdot \| _{\infty}$.
\end{dfn}
${}^{*} \mathbf{D} ={}^{*}[r_\mathbf{D},R_\mathbf{D}]$ is the ${}^{*}$-image of $\mathbf{D}$, and as such it is {\em internal}. \\
It is important to note that $st : {}^{*} \Omega \rightarrow \Omega$ is the standard part map, and $st(\omega)$ will be referred to as the {\em standard part} of $\omega$, for every $\omega \in  {}^{*} \Omega$. ${}^{\circ} z$ denotes the standard part
of a hyperreal $z$. \\

\begin{dfn}
	\label{neardefinition}

	For every $\omega \in \Omega$, if there exists $\widetilde{\omega} \in {}^{*}\Omega$ such that ${\| \widetilde{\omega} - {}^{*}\omega \|_{\infty} \simeq 0}$, then $\widetilde{\omega}$ is a {\em nearstandard point} in ${}^{*} \Omega$. This will be denoted as
	$ns(\widetilde{\omega}) \in  {}^{*}\Omega$.
\end{dfn}

For all hypernatural $N,$ let
\begin{equation}
\mathcal{L} _{N} = \left\lbrace \frac{K}{N\sqrt{N}}, \quad -N^2 \sqrt{R_{\mathbf{D}}} \leq K \leq N^2 \sqrt{R_{\mathbf{D}}}, \quad K \in {}^{*} \mathbb{Z} \right\rbrace,
\end{equation}
and the hyperfinite timelime
\begin{equation}
\mathbb{T}  = \left\lbrace 0, \frac{T}{N}, \cdots, -\frac{T}{N}+T, T  \right\rbrace.
\end{equation}
We consider $\mathcal{L}^{\mathbb{T}} _{N}$ as the canonical space of paths on the hyperfinite timeline, and $X^N = {(X^N_k)}_{k=0}^{N}$ as
the canonical process denoted by $X^N_k(\bar{\omega})= \bar{\omega}_k$ for ${\bar{\omega} \in \mathcal{L}^{\mathbb{T}} _{N}}$. $\mathcal{F}^N$ is the internal filtration generated by $X^N$.
The linear interpolation operator can be written as
$$\text{  }\widetilde{  } \text{ }: \text{ }\widehat{\cdot} \text{ } \circ \text{  } \iota ^{-1} \rightarrow {}^{*} \Omega , \quad \text{for  } \widetilde{\mathcal{L}^{\mathbb{T}} _{N}}\subseteq {}^{*} \Omega,$$
where $$\widehat{\omega}(t) := (\lfloor Nt/T \rfloor +1 -Nt/T)\omega_{ \lfloor Nt/T \rfloor } + (Nt/T - \lfloor Nt/T \rfloor)\omega_{\lfloor Nt/T \rfloor +1},$$
for $\omega \in \mathcal{L}^{N+1} _{N}$ and for all $t \in {}^*[0,T]$. $\lfloor y \rfloor$ denotes the greatest integer less than or equal to $y$ and $\iota : \mathbb{T} \rightarrow \{0, \cdots, N\}$ for $\iota : t \mapsto Nt/T$.

For the hyperfinite strong formulation of the volatility uncertainty, fix $N \in { }^{*}\mathbb{N} \setminus \mathbb{N}$. Consider $\left\lbrace \pm \frac{1}{\sqrt{N}} \right\rbrace ^{\mathbb{T}},$ and let $P_N$
be the uniform counting measure on $\left\lbrace \pm \frac{1}{\sqrt{N}} \right\rbrace ^{\mathbb{T}}$. $P_N$ can also be seen as a measure on $\mathcal{L}^{\mathbb{T}} _{N}$, concentrated on $\left\lbrace \pm \frac{1}{\sqrt{N}}\right\rbrace ^{\mathbb{T}}$.
Let ${\Omega _{N} = \{ \underline{\omega} = (\underline{\omega}_1, \cdots, \underline{\omega}_N); \underline{\omega}_i = \{\pm 1\}, i =1, \cdots
	, N\}}$, and let $\Xi_1, \cdots, \Xi_N$ be a ${}^{*}$-independent sequence of $\{\pm 1 \}$-valued random variables on $\Omega _{N}$
and the components of $\Xi _k$ are orthonormal in $L^2(P_N)$. We denote the hyperfinite random walk by
$$\mathbb{X}_t = \frac{1}{\sqrt{N}} \sum_{l=1}^{ Nt/T} \Xi _l \quad \text{ for all } t \in \mathbb{T}  .$$
The hyperfinite-time stochastic integral of some $F:\mathbb{T} \times \mathcal{L}^{\mathbb{T}} _{N} \rightarrow {}^* \mathbb{R}$ with respect to the hyperfinite random walk is given by
$$\sum_{s=0}^{t} F(s,\mathbb{X}) \Delta \mathbb{X}_s: \Omega _{N} \rightarrow {}^{*} \mathbb{R}, \quad \underline{\omega} \in \Omega _{N} \mapsto \sum_{s=0}^{t} F(s,\mathbb{X}(\underline{\omega})) \Delta \mathbb{X}_s(\underline{\omega}).$$
Thus, the hyperfinite set of martingale laws can be defined by
\begin{align}
\label{disc-meas}
\bar{\mathcal{Q}}_{\mathbf{D}^{\prime}_N}^{N} = \left\{ \begin{array}{lr}   P_N  \circ ( M^{F,\mathbb{X} })^{-1}; \text{  } F: \mathbb{T} \times \mathcal{L}^{\mathbb{T}}_{N} \rightarrow \sqrt{\mathbf{D}^{\prime}_{N}} \nonumber
\end{array}  \right\}
\end{align}
where
$${\mathbf{D}^{\prime}_N} = {}^{*}\mathbf{D} \cap \left( \frac{1}{N} {}^*\mathbb{N} \right)^2$$
and
$$M^{F, \mathbb{X}} = \left( \sum _{s=0}^{t} F (s, \mathbb{X})\Delta \mathbb{X}_ s \right) _{t\in \mathbb{T}}.$$

\begin{rem}
	Up to scaling,  $\bar{\mathcal{Q}}_{\mathbf{D}^{\prime}_N}^{N} = \mathcal{Q}_{\mathbf{D}^{\prime}_n}^n$.
\end{rem}

\subsection{Results and proofs}
\label{resultchap}

\begin{dfn}[(Uniform lifting of $\xi$)]
	\label{dfn-lifting}

	Let $\Xi: \mathcal{L}^{\mathbb{T}} _{N} \rightarrow {}^{*} \mathbb{R}$ be an internal function, and let $\xi: \Omega \rightarrow \mathbb{R}$ be a continuous function.  $\Xi$ is said to be a {\em uniform lifting} of $\xi$ if and only if
	$$\forall\bar{\omega} \in  \mathcal{L}^{\mathbb{T}} _{N} \Big(\widetilde{\bar{\omega}} \in ns({}^{*} \Omega) \Rightarrow {}^{\circ} \Xi (\bar{\omega}) = \xi (st(\widetilde{\bar{\omega}} )) \Big),$$
	where $st(\widetilde{\bar{\omega}})$ is defined
	with respect to the topology of uniform convergence on $\Omega$.
\end{dfn}

In order to construct the hyperfinite version of the $G$-expectation, we need to show that the ${ }^*$-image of $\xi$, ${}^{*}\xi$, with respect to $\widetilde{\bar{\omega}} \in ns( {}^{*} \Omega)$, is the canonical lifting of $\xi$
with respect to $st(\widetilde{\bar{\omega}}) \in \Omega$. i.e., for every $\widetilde{\bar{\omega}} \in ns( {}^{*} \Omega)$, ${}^{\circ} \left({}^{*}\xi (\widetilde{\bar{\omega}})\right)  = \xi (st(\widetilde{\bar{\omega}}))$. To do this, we need to show that ${ }^{*} \xi$ is S-continuous in every nearstandard point $\widetilde{\bar{\omega}}$.

It is easy to prove that there are two equivalent characteristics of $S$-continuity on ${}^* \Omega$.
\begin{rem}
	\label{s-contr}
	The following are equivalent for an internal function ${\Phi: { }^{*}\Omega \rightarrow { }^{*} \mathbb{R}}$;
	\begin{itemize}
		\label{s-cont}
		\item[$(1)$] $\forall \omega^{'} \in { }^{*}\Omega \left({ }^{*} \| \omega - \omega^{'} \|_{\infty} \simeq 0 \Rightarrow  { }^{*} |\Phi (\omega)- \Phi (\omega ^{'})|\simeq 0\right).$
		\item[$(2)$] $\forall \varepsilon \gg 0, \exists \delta \gg 0: \forall \omega^{'} \in { }^{*}\Omega \left( { }^{*} \| \omega - \omega^{'} \|_{\infty} < \delta \Rightarrow { }^{*} |\Phi (\omega)-  \Phi (\omega ^{'})| < \varepsilon \right).
		$
	\end{itemize}
\end{rem}
(The case of Remark \ref{s-cont} where $\Omega = \mathbb{R}$ is well known and proved in \citet[Theorem $5.1.1$]{Stroyan})

\begin{dfn}
	\label{s-contd}

	Let ${\Phi: { }^{*}\Omega \rightarrow { }^{*} \mathbb{R}}$ be an internal function. We say $\Phi$ is {\em$S$-continuous} in $\omega \in { }^{*}\Omega$, if and only if it satisfies one of the two equivalent conditions of Remark \ref{s-contr}.
\end{dfn}

\begin{pro}
	\label{s-pro}

	If ${\xi: \Omega \rightarrow \mathbb{R}}$ is a continuous function satisfying $|\xi (\omega)| \leq a(1+ \|\omega \|_{\infty})^b$, for $a,b>0$, then, $\Xi= {}^{*}\xi \circ \widetilde{\cdot}$ is a uniform lifting of $\xi$.
\end{pro}

\begin{proof}
	Fix $\omega \in \Omega$. By definition, $\xi$ is continuous on $\Omega$. i.e., for all $\omega \in \Omega$, and for every $ \varepsilon \gg 0$, there is a $\delta \gg 0 $, such that for every $ \omega^{'} \in \Omega$, if
	\begin{equation}
	\label{xi-cont}
	\| \omega - \omega^{'} \|_{\infty} <  \delta , \text{ then } |\xi (\omega) - \xi (\omega^{'})| <  \varepsilon.
	\end{equation}
	By the Transfer Principle: For all $\omega \in \Omega$, and for every $ \varepsilon \gg 0$, there is a $\delta \gg 0 $, such that for every $ \omega^{'} \in {}^{*} \Omega$, \eqref{xi-cont} becomes,
	\begin{equation}
	\label{s-del}
	{}^{*} \|{}^{*} \omega - \omega^{'} \|_{\infty} <  \delta , \text{ and } {}^{*} | {}^{*} \xi ({}^{*}\omega) - {}^{*}  \xi (\omega^{'})| <  \varepsilon.
	\end{equation}
	So, ${}^{*} \xi$ is $S$-continuous in ${}^{*} \omega$ for all $\omega \in \Omega$. Applying the equivalent characterization of $S$-continuity, Remark \ref{s-contr}, \eqref{s-del} can be written as \
	\begin{equation*}
	{ }^{*} \| { }^{*} \omega - \omega^{'} \|_{\infty} \simeq 0, \text{ and } { }^{*} |{ }^{*} \xi ({ }^{*} \omega) - { }^{*} \xi (\omega^{'})| \simeq 0.
	\end{equation*}
	We assume $\widetilde{\bar{\omega}}$ to be a nearstandard point. By Definition \ref{neardefinition}, this simply implies,
	\begin{equation}
	\label{near}
	\forall \widetilde{\bar{\omega}} \in ns( {}^{*} \Omega), \text{  } \exists \omega \in \Omega :  { }^{*} \| \widetilde{\bar{\omega}} -  { }^{*}\omega  \|_{\infty} \simeq 0.
	\end{equation}
	Thus, by $S$-continuity of ${ }^{*} \xi$ in ${}^{*} \omega$,\\
	$${ }^{*} |{ }^{*} \xi (\widetilde{\bar{\omega}})    - { }^{*} \xi ({ }^{*} \omega)| \simeq 0. $$
	Using the triangle inequality, if $\omega^{'}  \in {}^{*} \Omega$ with ${}^*\|   \widetilde{\bar{\omega}}  - \omega^{'} \|_{\infty}\simeq 0 $,
	$${ }^{*} \|{ }^{*}\omega  - \omega^{'} \|_{\infty} \leq   { }^{*} \|  { }^{*}\omega - \widetilde{\bar{\omega}} \|_{\infty} + { }^{*}\| \widetilde{\bar{\omega}} - \omega^{'} \|_{\infty} \simeq 0$$
	and therefore again by the $S$-continuity of ${ }^{*} \xi$ in $ { }^{*} \omega$,
	$${ }^{*} |{ }^{*} \xi ({}^{*} \omega)    - { }^{*} \xi (\omega ^{'})|  \simeq 0 .$$
	And so,
	\begin{align*}
	{ }^{*} |{ }^{*} \xi (\widetilde{\bar{\omega}})- { }^{*} \xi (\omega ^{'})| & \leq { }^{*} |{ }^{*} \xi (\widetilde{\bar{\omega}}) - { }^{*} \xi ({ }^{*} \omega)| + { }^{*} |{ }^{*} \xi ({ }^{*}\omega)- { }^{*} \xi (\omega^{'})|
	\simeq 0. &
	\end{align*}
	Thus, for all $ \widetilde{\bar{\omega}} \in ns( {}^{*} \Omega)$ and $ \omega^{'} \in {}^{*} \Omega  $, if ${ }^{*} \| \widetilde{\bar{\omega}} - \omega^{'} \|_{\infty} \simeq 0$, then,
	\begin{equation*}
	{ }^{*} |{ }^{*} \xi (\widetilde{\bar{\omega}}) - { }^{*} \xi (\omega ^{'})|\simeq 0.
	\end{equation*}
	Hence, ${ }^{*} \xi$ is S-continuous in $\widetilde{\bar{\omega}}$. Equation \eqref{near} also implies
	$$\widetilde{\bar{\omega}} \in m (\omega) \left( m (\omega) = \bigcap \{{ }^{*} \mathcal{O}; \mathcal{O} \text{ is an open neighbourhood of }\omega \}\right)$$
	such that $\omega$ is unique, and in this case $st(\widetilde{\bar{\omega}} ) = \omega$.\\
	Therefore,
	\begin{equation*}
	{}^{\circ} \Big({}^{*}\xi (\widetilde{\bar{\omega}})\Big)  = \xi (st(\widetilde{\bar{\omega}})).
	\end{equation*}
\end{proof}

\begin{dfn}

	Let $\bar{\mathcal{E}}: {{ }^{*} \mathbb{R}}^{\mathcal{L}_{N}^{\mathbb{T}}} \rightarrow { }^{*} \mathbb{R}$. We say that $\bar{\mathcal{E}}$ lifts $\mathcal{E}^G$ if and only if for every $\xi:\Omega \rightarrow \mathbb{R}$ that satisfies $|\xi (\omega)| \leq a(1+ \|\omega \|_{\infty})^b$ for some $a, b> 0$,
	$$\bar{\mathcal{E}}({ }^{*}\xi \circ \tilde{\cdot})  \simeq  \mathcal{E}^G (\xi). $$
\end{dfn}

\begin{thm}
	\begin{equation}
	\label{G-thm}
	\max _{\bar{Q}\in \bar{\mathcal{Q}}^N_{\mathbf{D}^{\prime}_N}} \mathbb{E}^{\bar{Q}}[\cdot] \text{  lifts  }\mathcal{E}^G(\xi) .
	\end{equation}
\end{thm}

\begin{proof}
	From Theorem \ref{max-result},
	\begin{equation}
	\label{stand-G}
	\max_{\mathbb{Q} \in \mathcal{Q}_{\mathbf{D}^{\prime}_n}^{n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)] \rightarrow \mathcal{E}^G(\xi), \quad \text{as } n\rightarrow \infty  .
	\end{equation}
	For all $N \in { }^{*}\mathbb{N} \setminus \mathbb{N},$ we know that \eqref{stand-G} holds if and only if
	\begin{equation}
	\label{stand-G2}
	\max _{Q\in { }^{*}\mathcal{Q}^N_{\mathbf{D}^{\prime}_N}} \mathbb{E}^Q [{ }^{*} \xi (\widehat{X}^N)]\simeq \mathcal{E}^G (\xi),
	\end{equation}
	(see \citet{Albeverio}, Proposition $1.3.1$). Now, we want to express \eqref{stand-G2} in term of $\bar{\mathcal{Q}}_{\mathbf{\mathbf{D}^{\prime}_N}}^{N}$. i.e., to show that
	\begin{equation*}
	\max _{ \bar{Q} \in \bar{\mathcal{Q}}^N_{\mathbf{D}^{\prime}_N}} \mathbb{E}^{\bar{Q}}[{ }^{*}\xi \circ \tilde{\cdot}]\simeq \mathcal{E}^G(\xi) .
	\end{equation*}
	To do this, use
	$$\mathbb{E}^Q [{ }^{*} \xi \circ \hat{ \cdot}] = \mathbb{E}^Q [{ }^{*} \xi \circ \hat{\cdot} \circ \iota ^{-1} \circ \iota]$$
	and
	\begin{align*}
	\label{tran-measure}
	\mathbb{E}^Q [{ }^{*} \xi \circ \hat{\cdot} \circ \iota ^{-1} \circ \iota]& = \mathbb{E}^Q [{ }^{*} \xi \circ \tilde{\cdot} \circ \iota] \\ \nonumber
	& = \int _{{ }^{*}\mathbb{R}^{N+1}}{ }^{*} \xi \circ \tilde{\cdot} \circ \iota dQ, \quad \text{(transforming measure)}\\ \nonumber
	& = \int _{{ }^{*}\mathbb{R}^{\mathbb{T}}}{ }^{*} \xi \circ \tilde{\cdot} d(Q \circ j), \\ \nonumber
	& = \mathbb{E}^{Q \circ j} [{ }^{*} \xi \circ \tilde{\cdot}]
	\end{align*}
	for $j : { }^{*}\mathbb{R}^{\mathbb{T}} \rightarrow { }^{*}\mathbb{R}^{N+1}$, $(xt)_{t \in \mathbb{T}} \mapsto\left( \frac{xNt}{T} \right)_{t \in \mathbb{R}^{N+1}}.$\\
	Thus, $$\bar{\mathcal{Q}}^N_{\mathbf{D}^{\prime}_N} = \{ Q \circ j : Q \in { }^{*}\mathcal{Q}^N_{\mathbf{D}^{\prime}_N} \} .$$
	This implies,
	$$\max _{ \bar{Q} \in \bar{\mathcal{Q}}^N_{\mathbf{D}^{\prime}_N}} \mathbb{E}^{\bar{Q}}[{ }^{*}\xi \circ \tilde{\cdot}] = \max _{Q\in { }^{*}\mathcal{Q}^N_{\mathbf{D}^{\prime}_N}} \mathbb{E}^Q [{ }^{*} \xi \circ \hat{\cdot}].$$
\end{proof}

\section*{Appendix}

\begin{proof}[Proof of Lemma \ref{equivmeas}]
	From the above equation, we can say that $\Delta M^f_{k} = f(k,\mathbb{X})\xi_k$. And by the orthonormality property of $\xi_k$, we have
	$$\mathbb{E}^{P_n}[f(k,\mathbb{X})^2\xi_k^2|\mathcal{F}^n_{k}] = \mathbb{E}^{P_n}[f(k,\mathbb{X})^2| \mathcal{F}^n_{k}] \leq \mathbb{E}^{P_n}[(\sqrt{R_{\mathbf{D}}})^2|\mathcal{F}^n_{k}] = R_{\mathbf{D}} \quad P_n\text{ a.s.},$$
	as $|\xi_k|=1$, $f(\cdots)^2\in \mathbf{D}$ implies
	$$| (\Delta M_{k}^f)^2 | = |f(k,\mathbb{X})|^2 \in [r_{\mathbf{D}}, R_{\mathbf{D}}]\quad P_n\text{ a.s}.$$
\end{proof}

\subsection*{Density argument verification}
Let
$$f: \xi \mapsto \sup_{P \in \mathcal{Q}_\mathbf{D}} \mathbb{E}^{P} [\xi] $$
and
$$g: \xi \mapsto  \lim_{n \rightarrow \infty} \sup_{\mathbb{Q}\in \mathcal{Q}^{n}_{{\mathbf{D}^{\prime}_n}/n}} \mathbb{E}^{\mathbb{Q}} [\xi (\widehat{X}^n)].$$
From \eqref{final-equation}, we know that for all $\xi \in \mathcal{C}_{b}(\Omega, \mathbb{R}), \text{ } f(\xi) = g(\xi).$
Since $\mathbb{L}^{1}_{*}$ is the completion of $\mathcal{C}_{b}(\Omega, \mathbb{R})$ under the norm $\| \cdot \|_{*}$,  $\mathcal{C}_{b}(\Omega, \mathbb{R})$
is dense in $\mathbb{L}^{1}_{*};$ and we want to prove for all $\xi \in \mathbb{L}^{1}_{*}$, $f(\xi) = g(\xi).$ To prove this, it is sufficient to show that $f$ and $g$
are continuous with respect to the norm $\| \cdot \| _{*}.$

\subsubsection*{For continuity of $f$:}

For all $ P \in \mathcal{Q}_\mathbf{D}$ and $\xi, \xi^{'} \in \mathbb{L}^{1}_{*}$,

$$\sup _{ P \in \mathcal{Q}_\mathbf{D}}\mathbb{E}^{P} [\xi] - \sup_{ P \in \mathcal{Q}_\mathbf{D}} \mathbb{E}^{P} [\xi ^{'}] \leq \sup _{ P \in \mathcal{Q}_\mathbf{D}}\mathbb{E}^{P} [| \xi - \xi^{'} |].$$
Since, $\mathcal{Q}_\mathbf{D} \subseteq \mathcal{Q}$,
\begin{equation}
\label{ap-conf}
\sup _{ P \in \mathcal{Q}_\mathbf{D}}\mathbb{E}^{P} [\xi] - \sup_{ P \in \mathcal{Q}_\mathbf{D}} \mathbb{E}^{P} [\xi ^{'}] \leq \| \xi - \xi ^{'} \| _{*}.
\end{equation}
Interchanging $\xi$ and $\xi^{'}$,

\begin{equation}
\label{ap-conf1}
\sup _{ P \in \mathcal{Q}_\mathbf{D}}\mathbb{E}^{P} [\xi ^{'}] - \sup_{ P \in \mathcal{Q}_\mathbf{D}} \mathbb{E}^{P} [\xi] \leq \| \xi ^{'} - \xi  \| _{*}.
\end{equation}
Adding \eqref{ap-conf} and \eqref{ap-conf1}, we have
$|f(\xi) - f(\xi ^{'})  | \leq  \| \xi  - \xi ^{'} \| _{*}. $
\subsubsection*{For continuity of $g$:}
We follow the same argument as above.

\begin{proof}[Proof of Remark \ref{s-contr}]
	Let $\Phi$ be an internal function such that condition $(1)$ holds. To show that $(1) \Rightarrow (2)$, fix $\varepsilon \gg 0$. We shall show there exists a $\delta$ for
	this $\varepsilon$ as in condition $(2)$. Since $\Phi$ is internal, the set
	$$I = \left\lbrace \delta \in  { }^{*} \mathbb{R}_{>0}: \text{ }\forall \omega^{'} \in { }^{*}\Omega \text{ } ( { }^{*} \| \omega - \omega^{'} \|_{\infty} < \delta \Rightarrow { }^{*} |\Phi (\omega)-  \Phi (\omega ^{'})| < \varepsilon ) \right\rbrace,$$
	is internal by the Internal Definition Principle and also contains every positive infinitesimal. By Overspill (cf. \citet[Proposition $1.27$]{Albeverio}) $I$ must then contain some positive $\delta \in \mathbb{R}$.\\
	Conversely, suppose condition (1) does not hold, that is, there exists some $\omega^{'} \in { }^{*}\Omega$ such that
	$${ }^{*} \| \omega - \omega^{'} \|_{\infty} \simeq 0 \text{ and }  { }^{*} |\Phi (\omega)- \Phi (\omega ^{'})| \text{ is not infinitesimal}.$$
	If $\varepsilon = \min(1, { }^{*}| \Phi (\omega)- \Phi (\omega ^{'})|/2),$ we know that for each standard $\delta >0,$ there is a point $\omega ^{'}$ within $\delta$ of $\omega$ at which $\Phi (\omega ^{'})$ is farther than $\varepsilon$
	from $\Phi (\omega)$. This shows that condition $(2)$ cannot hold either.
\end{proof}