text:
\section{Introduction}
A well-known chestnut presents a sequence of arcs $\gamma_n\colon[0,1]\to\numbersystem{R}^2$ between fixed points $a$ and $b$ at distance $2$, each consisting of $2^{n-1}$ semicircular arcs of radius $2^{n-1}$, converging (uniformly) to the straight-line arc $\gamma_0\colon[0,1]\to\numbersystem{R}^2$, $\gamma_0(t)=(1-t)a+tb$ (Fig.~\ref{fig:circle}).
\begin{figure}[t]
\centering
\begin{tikzpicture}[thick, scale=3]
    \draw (0,0) node [below] {$2$};
    \draw (1,0) arc (0:180:1cm);
    \draw (45:1cm) node[above right] {$\pi$};
    \draw[black] (1,0) \foreach \x in {1,2} {arc (0:180:0.5cm)};
\draw[black!85] (1,0) \foreach \x in {1,...,4} {arc (0:180:0.25cm)};
\draw[black!70] (1,0) \foreach \x in {1,...,8} {arc (0:180:0.125cm)};
\draw[black!55] (1,0) \foreach \x in {1,...,16} {arc (0:180:0.0625cm)};
\draw[black!40] (1,0) \foreach \x in {1,...,32} {arc (0:180:0.03125cm)};
\draw (-1,0) node[left] {$a$} -- (1,0) node[right] {$b$};
\end{tikzpicture}
\caption{A sequence of arcs, each of length $\pi$, converging to a segment of length $2$.}\label{fig:circle}
\end{figure}
That the length of each $\gamma_n$ is $\pi$ and of the limit curve is $2$ shows that arc length is not a continuous functional.
Instead, for any metric space $(X,d)$, arc length is a lower semicontinuous functional on the space $C([0,1],X)$ of continuous functions with the supremum norm.
This is most easily seen by noting that arc length can be defined as the supremum of the polygonal approximation functionals
\begin{equation}\label{eq:arclength}
\gamma\mapsto\sum_{i=1}^{n-1} d(\gamma(a_i),\gamma(a_{i+1})),
\end{equation}
where $0\leq a_1<\dots<a_n\leq 1$, each one of which is continuous, and applying the result that the supremum of a collection of lower semicontinuous functions is again lower semicontinuous.
Thus, the length of the limit arc can never be larger than the limit of the lengths of the arcs in the sequence, but can also be strictly smaller.

This observation also demonstrates that it is not completely straightforward that the Koch curve (Figure~\ref{koch0}) has infinite length, despite being the limit of curves $\kappa_n$ of length $(4/3)^n$, tending to infinity.
\begin{figure}
\centering
\begin{tikzpicture}[thin,decoration=Koch snowflake, scale=5]
    \draw[black!20] decorate{ (-1,0) -- (1,0) };
    \draw[black!30] decorate{ decorate{ (-1,0) -- (1,0) }};
    \draw[black!40] decorate{ decorate{ decorate{ (-1,0) -- (1,0) }}};
    \draw[thin,black!60] decorate{ decorate{ decorate{ decorate{ (-1,0) -- (1,0) }}}};
    \draw[thin,black!80] decorate{ decorate{ decorate{ decorate{ decorate{ (-1,0) -- (1,0) }}}}};
    \draw[thin,black] decorate{ decorate{ decorate{ decorate{ decorate{ decorate{ (-1,0) -- (1,0) }}}}}};
\end{tikzpicture}
\caption{The Koch curve}\label{koch0}
\end{figure}
This point is usually glossed over in discussions of the Koch curve, including the original paper of Helge von Koch \cite{Koch1904}.
In Figure~\ref{koch2} we obtain $\delta_n$ by shrinking $\kappa_n$ by a linear factor of $(4/5)^{n-1}$ and adding segments at the two endpoints of the shrunken curve so that the endpoints are again at the same distance as before.
\begin{figure}
\centering
\begin{tikzpicture}[thin,decoration=Koch snowflake, scale=2.8]
    \draw decorate{ (-1,0) -- (1,0) } node[right=2mm] {$\delta_1$};

    \tikzmath{
        real \a, \b, \c;
        \a = 0.8;
        \b = \a;
        \c =-\b;
    }
    \draw[yshift=-0.4cm] (-1,0) -- (\c,0) decorate{ decorate{ -- (\b,0) }} -- (1,0) node[right=2mm] {$\delta_2$};

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \begin{scope}[yshift=-0.72cm]
        \draw (-1,0) -- (\c,0);
        \draw decorate{ decorate{ decorate{ (\c,0) -- (\b,0) }}};
        \draw (\b,0) -- (1,0) node[right=2mm] {$\delta_3$};
    \end{scope}

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \draw[yshift=-1cm] (-1,0) -- (\c,0) decorate{ decorate{ decorate{ decorate{ -- (\b,0) }}}} -- (1,0) node[right=2mm] {$\delta_4$};

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \begin{scope}[yshift=-1.25cm]
        \draw (-1,0) -- (\c,0);
        \draw[thin] decorate{ decorate{ decorate{ decorate{ decorate{ (\c,0) -- (\b,0) }}}}};
        \draw (\b,0) -- (1,0) node[right=2mm] {$\delta_5$};
    \end{scope}

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \begin{scope}[yshift=-1.5cm]
        \draw (-1,0) -- (\c,0);
        \draw[thin] decorate{ decorate{ decorate{ decorate{ decorate{ (\c,0) -- (\b,0) }}}}};
        \draw (\b,0) -- (1,0) node[right=2mm] {$\delta_6$};
    \end{scope}

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \begin{scope}[yshift=-1.7cm]
        \draw (-1,0) -- (\c,0);
        \draw[thin] decorate{ decorate{ decorate{ decorate{ decorate{ (\c,0) -- (\b,0) }}}}};
        \draw (\b,0) -- (1,0) node[right=2mm] {$\delta_7$};
    \end{scope}

    \tikzmath{
        \b = \b * \a;
        \c =-\b;
    }
    \begin{scope}[yshift=-1.9cm]
        \draw (-1,0) -- (\c,0);
        \draw[thin] decorate{ decorate{ decorate{ decorate{ decorate{ (\c,0) -- (\b,0) }}}}};
        \draw (\b,0) -- (1,0) node[right=2mm] {$\delta_8$};
    \end{scope}

\end{tikzpicture}
\caption{Sequence of arcs of length $>(16/15)^n$ converging to a segment}\label{koch2}
\end{figure}
Then the length of $\delta_n$ is bounded from below by $(16/15)^{n}$, but the limit curve will be the segment $\gamma_0$.
In order to show that the Koch curve has infinite length, one has to observe that the vertices of each polygonal arc $\kappa_n$ occurs in all subsequent $\kappa_m$, $m>n$, hence also in the limit curve, and therefore by the elementary definition of arc length as the supremum of \eqref{eq:arclength}, the length of the Koch curve is bounded from below by the length of each $\kappa_n$.

The lower semicontinuity of arc length was generalised in the early $20$th century from arcs to \emph{continua}, that is, connected compact subsets of a metric space.
The first to publish a proof of the lower semicontinuity of the length of continua seems to be Stanis\l{}aw Go\l\k{a}b \cite{Golab1928}, and this theorem is nowadays known as Go\l\k{a}b's Theorem.
Recently, it has been used and generalised in existence proofs in the Calculus of Variations \cite{Dalmaso1992, DalmasoToader2002, PaoliniStepanov2013}.
See Section~\ref{history} for further historical remarks.

The main purpose of this paper is to give a simple, self-contained proof of Go\l\k{a}b's Theorem by using an almost forgotten definition of the length of continua due to Karl Menger \cite{Menger1931} and Gustave Choquet \cite{Choquet1938}.
This proof uses no measure theory apart from the definition of outer linear measure, and no results on the structure of continua.

\section{Five formulations of lower semicontinuity}
Let $(X,d)$ be a metric space.
Denote the open ball with centre $x\in X$ and radius $r>0$ by $B(x,r)=\setbuilder{y\in X}{d(x,y)<r}$ and
the power set of $X$ by
$\mathscr{P}(X) = \setbuilder{A}{A\subseteq X}$.
For convenience we define $\sup\emptyset=0$ and $\inf\emptyset=\infty$.
Define the \emph{diameter} of $A\subseteq X$ as\[\diam(A)=\sup\setbuilder{d(a_1,a_2)}{a_1,a_2\in A}.\]
This gives an extended real-valued function $\diam\colon\mathscr{P}(X)\to[0,\infty]$.

The \emph{outer linear measure} $L^*\colon\mathscr{P}(X)\to[0,\infty]$ (also called $1$-dimensional outer Hausdorff measure) is defined as follows.
A countable family $\setbuilder{U_i}{i\in\numbersystem{N}}$ of subsets of $X$ is called a \emph{$\delta$-cover} of $A$ if $\diam(U_i)\leq\delta$ for all $i\in\numbersystem{N}$ and $A\subseteq\bigcup_{i\in\numbersystem{N}}U_i$.
For any $\delta>0$, let \[L^*_\delta(A)=\inf\setbuilder{\sum_{i\in\numbersystem{N}} \diam(U_i)}{\text{$\{U_i:i\in\numbersystem{N}\}$ is a $\delta$-cover of $A$}},\]
and let $L^*(A)=\sup\setbuilder{L^*_\delta(A)}{\delta>0} = \lim_{\delta\to 0}L^*_\delta(A)$.
Note that $L^*$ is \emph{monotone}: if $A\subseteq B$ then $L^*(A)\leq L^*(B)$.

Roughly, Go\l\k{a}b's theorem \cite{Golab1928} asserts that $L^*$ is lower semicontinuous on the collection of continua of $X$ with a suitable topology.
Instead of starting off with a discussion of these topologies, we give five concrete formulations of Go\l\k{a}b's Theorem, four of which can be found in the literature.
After giving a self-contained proof of the strongest of these, we discuss the topologies behind the different formulations.

We need the following definitions.
For $a\in X$ and $A,B\subseteq X$, define the \emph{distance from $a$ to $B$} as \[d(a,B)=\inf\setbuilder{d(a,b)}{b\in B},\]
the \emph{excess} (or \emph{Hausdorff hemimetric}) \emph{of $A$ over $B$} as \[e(A,B)=\sup\setbuilder{d(a,B)}{a\in A},\]
and the \emph{Hausdorff distance between $A$ and $B$} as \[D(A,B)=\max\set{e(A,B),e(B,A)}.\]
These again define extended real-valued functions $d\colon X\times\mathscr{P}(X)\to[0,\infty]$, $e\colon \mathscr{P}(X)^2\to[0,\infty]$ and $D\colon\mathscr{P}(X)^2\to[0,\infty]$.
Note that, according to our definitions, $d(a,\emptyset)=e(A,\emptyset)=D(A,\emptyset)=\infty$ and $e(\emptyset,B)=0$.
A (metric) \emph{continuum} is a connected, compact metric space.

The following theorem was proved by Stanis\l{}aw Go\l\k{a}b in 1928 \cite{Golab1928}.
\begin{golabtheorem}
Let $A$ be a continuum and $\netbuilder{A_n}{n\in\numbersystem{N}}$ a sequence of continua in the metric space $X$ such that $\lim_{n\to\infty} D(A,A_n)=0$.
Then \[L^*(A)\leq\liminf\limits_{n\to\infty}L^*(A_n).\]
\end{golabtheorem}
There are various ways to strengthen this theorem.
It is not necessary to assume that $A$ is a continuum, as this follows from the convergence of $A_n$ to $A$ with respect to Hausdorff distance \cite[Theorem~3.18]{Falconer1986}.
Although Go\l\k{a}b's Theorem is usually formulated in terms of Hausdorff distance, Go\l\k{a}b already remarked that the theorem still holds if we only assume that $\lim_{n\to\infty} e(A,A_n)=0$.
However, this weaker assumption does not imply that $A$ is a continuum, as $\lim_{n\to\infty} e(A',A_n)=0$ holds for any $A'\subseteq A$.
This does not matter, as the compactness of $A$ or the $A_n$ is not essential for the conclusion to hold.
We thus state the following version of Go\l\k{a}b's Theorem.
\begin{gtheorem}\label{golab}
Let $\netbuilder{A_n}{n\in\numbersystem{N}}$ be a sequence of connected subsets of the metric space $X$ and $A\subseteq X$ such that $\lim_{n\to\infty} e(A,A_n)=0$.
Then \[L^*(A)\leq\liminf\limits_{n\to\infty}L^*(A_n).\]
\end{gtheorem}
Independently of Go\l\k{a}b, in 1936 Orrin Frink \cite{Frink1936} proved the lower semicontinuity of $L^*$ on continua in the following form.
Although he proved only the first statement, the second statement follows in a very similar way.
Again we formulate it for connected sets.
\begin{gtheorem}\label{frink}
Let $A$ be a subset of the metric space $X$.
\begin{enumerate}
\item If $L^*(A)<\infty$ then for all $\varepsilon>0$ there exists $\delta>0$ such that for any connected $B\subseteq X$ with $e(A,B)<\delta$, $L^*(A)<L^*(B)+\varepsilon$.
\item If $L^*(A)=\infty$, then for all $M>0$ there exists $\delta>0$ such that for any connected $B\subseteq X$ with $e(A,B)<\delta$, $L^*(B)>M$.
\end{enumerate}
\end{gtheorem}
Theorem~\ref{golab} follows easily from Theorem~\ref{frink}.
Indeed, if $L^*(A)=\infty$, then Theorem~\ref{frink} gives that for any $M>0$ there exists $n_0\in\numbersystem{N}$ such that for all $n\geq n_0$, $L^*(A_n)>M$, which implies $\liminf_{n\to\infty} L^*(A_n)=\infty$, and if $L^*(A)<\infty$, then Theorem~\ref{frink} gives that for any $\varepsilon>0$ there exists $n_0\in\numbersystem{N}$ such that for all $n\geq n_0$, $L^*(A)<L^*(A_n)+\varepsilon$, which implies $L^*(A)\leq\liminf_{n\to\infty} L^*(A_n)+\varepsilon$.

A third form of Go\l\k{a}b's Theorem may be found in a 1989 paper of M\'aty\'as Bogn\'ar \cite{Bognar1989}.
Although Bogn\'ar only formulated it for sequences, the generalisation to nets presents no problems.
Let $(\Lambda,\preccurlyeq)$ be a directed set and $\netbuilder{A_\lambda}{\lambda\in\Lambda}$ a net of subsets of $X$.
We say that $x\in X$ is a \emph{limit point} of $\net{A_\lambda}$ if for each $\varepsilon>0$ there exists $\lambda_0\in\Lambda$ such that for all $\lambda\succ \lambda_0$, $A_\lambda\cap B(x,\varepsilon)\neq\emptyset$.
Then the (Painlev\'e--Kuratowski) \emph{lower limit} of $\net{A_\lambda}$, denoted by $\li A_\lambda$, is defined to be the set  of all limit points of $\net{A_\lambda}$.
(The topology defined by this convergence is not necessarily first countable, hence the use of nets.)
\begin{gtheorem}\label{bognar}
Let $\net{A_\lambda}$ be a net of connected subsets of a metric space $X$.
Then \[L^*(\li A_\lambda)\leq\liminf_{\lambda}L^*(A_\lambda).\]
\end{gtheorem}
It is again easy to see that Theorem~\ref{bognar} implies Theorem~\ref{golab}.
Indeed, for any $\varepsilon>0$ there exists $n_0\in\numbersystem{N}$ such that $e(A,A_n)<\varepsilon$ for all $n\geq n_0$.
Thus for any $x\in A$ we have $d(x,A_n)<\varepsilon$, which implies that $x\in\li_n A_n$.
Then Theorem~\ref{bognar} and the monotonicity of $L^*$ gives that $L^*(A)\leq L^*(\li A_n)\leq\liminf_{n\to\infty} L^*(A_n)$.

A fourth form of Go\l\k{a}b's Theorem appears in a 1992 paper of David Fremlin \cite[Lemma~5C]{Fremlin1992}.
We say that a set $A$ \emph{hits} a collection $\mathscr{U}$ of sets if $A\cap U\neq\emptyset$ for all $U\in\mathscr{U}$.
\begin{gtheorem}\label{fremlin}
Let $A$ be a subset of the metric space $X$ and let $\alpha\geq 0$.
Suppose that for any finite collection $\mathscr{U}$ of open sets in $X$ which is hit by $A$, there exists a connected $C\subset X$ that hits $\mathscr{U}$, such that $L^*(C)\leq\alpha$.
Then $L^*(A)\leq\alpha$.
\end{gtheorem}
This theorem easily implies Theorem~\ref{bognar}.

We state a final version very close to Theorem~\ref{fremlin}, but in a form parallel to Theorem~\ref{frink}.
\begin{gtheorem}\label{lsc}
Let $A$ be a subset of a metric space $X$.
\begin{enumerate}
\item If $L^*(A)<\infty$ then for all $\varepsilon>0$ there exists a finite collection $\mathscr{U}$ of open sets in $X$ hit by $A$ such that for any connected $B\subseteq X$ that hits $\mathscr{U}$, $L^*(A)<L^*(B)+\varepsilon$.
\item If $L^*(A)=\infty$, then for all $M>0$ there exists a finite collection $\mathscr{U}$ of open sets in $X$ hit by $A$ such that for any connected $B\subseteq X$ that hits $\mathscr{U}$, $L^*(B)>M$.
\end{enumerate}
\end{gtheorem}
This theorem easily implies both Theorems~\ref{frink} and \ref{fremlin}.

In the next section we give a self-contained proof of Theorem~\ref{lsc}.
This is done in two steps.
First, we introduce the Menger--Choquet length of an arbitrary subset of a metric space and present a very simple proof of its semicontinuity formulated in a manner analogous to Theorem~\ref{lsc} (Proposition~\ref{MClsc}).
Then we state our main result (Theorem~\ref{maintheorem}), that the outer linear measure of a connected subset of a metric space $X$ equals its Menger--Choquet length.
This result generalises an old theorem of Choquet \cite{Choquet1938}.
Its proof is in Section~\ref{proof}.
Up to here everything is self-contained (using some elementary notions from graph theory).
In particular, everything is done without specifying exactly what we mean by lower semicontinuity.
In Section~\ref{topology} we discuss the two topological senses in which the above versions of Go\l\k{a}b's Theorem assert the lower semicontinuity of $L^*$ on connected sets or on continua.
Section~\ref{history} contains a discussion of the history and applications of Go\l\k{a}b's Theorem.

\section{Menger--Choquet length and Steiner trees}\label{tree}
It is clear that the outer linear measure of a set $A$, as defined in the previous section, is independent of the metric space of which $A$ is a subset.
The same will not be true for what we will call the Menger--Choquet length, introduced next.

As usual, we consider a \emph{graph} on a finite non-empty set of \emph{vertices} $V$ to be a pair $G=(V,E)$ where $E\subseteq\binom{V}{2}$, the set of \emph{edges} of $G$, is a set of unordered pairs of vertices.
The \emph{degree} of a vertex $x$ of $G$ is
$\deg_G(x) = \card{\setbuilder{e\in E}{x\in e}}$.
A \emph{path} with \emph{endpoints} $a$ and $b$ is a subgraph of $G$ with distinct vertices $v_1,\dots,v_n$, $n\geq 1$, where $a=v_1$, $b=v_n$, and edges $\set{v_i,v_{i+1}}\in E$ for $i=1,\dots,n-1$.
A graph $G$ is \emph{connected} if for any $a,b\in V$, $G$ contains a path with endpoints $a$ and $b$, and a \emph{tree} if there is a unique path between any two vertices.
A \emph{cycle} is a graph with distinct vertices $v_1,\dots,v_n$, $n\geq 3$, and edges $\set{v_i,v_{i+1}}$ for $i=1,\dots,n-1$ as well as $\set{v_1,v_n}$.
If the vertex set $V$ of a graph $G=(V,E)$ is contained in the metric space $(X,d)$, then the \emph{length} of $G$ is defined to be $\ell(G)=\sum_{\set{a,b}\in E}d(a,b)$.
Our graphs are not geometric in any sense: We do not require $a$ and $b$ to be joined by an arc in $X$ if $\set{a,b}\in E$.

For any finite non-empty set $P\subseteq X$, define
\newlength{\myl}
\settowidth{\myl}{$G=(V,E)$ is a connected graph}
\[\operatorname{smt}(P)=\inf\setbuilder{\ell(G)}{\text{$G=(V,E)$ is connected with $P\subseteq V\subseteq X$}}.\]
For example, if $X$ is the Euclidean plane and $P$ the vertex set of an equilateral triangle of side length $1$, then $\operatorname{smt}(P)=\sqrt{3}$, with a shortest graph connecting the vertices to the centroid of the triangle.
Since any connected $G=(V,E)$ contains a tree $T=(V,E')$ on the same vertex set (a \emph{spanning tree} of $G$), we may restrict the graphs in the definition of $\operatorname{smt}(P)$ to be trees.
Trees with a vertex set that contains $P$ are called \emph{Steiner trees} on $P$.
The vertices in $V\setminus P$ (if any) are the \emph{Steiner points} of the Steiner tree.
(If the infimum in the definition of $\operatorname{smt}(P)$ is attained, any tree $T$ with vertex set containing $P$ such that $\operatorname{smt}(P)=\ell(T)$ is called a \emph{Steiner minimal tree} of $P$.)
Without loss of generality, we may restrict the Steiner trees in the definition of $\operatorname{smt}(P)$ to the \emph{proper Steiner trees} on $P$, that is, Steiner trees on $P$ such that all Steiner points have degree at least $2$.
(We could even restrict the Steiner trees to those in which all Steiner points have degree at least $3$, but it will be important later to allow Steiner points of degree $2$.)
As observed by Gustave Choquet \cite{Choquet1938} in 1938, it is immediate that if $P$ is contained in the finite set $Q$, then $\operatorname{smt}(P)\leq\operatorname{smt}(Q)$.
For any $A\subseteq X$ we define its \emph{Menger--Choquet length} to be
\[ L_{MC}(A) = \sup\setbuilder{\operatorname{smt}(P)}{\text{$P$ is a finite non-empty subset of $A$}}.\]
This defines a monotone function $L_{MC}\colon\mathscr{P}(X)\to[0,\infty]$.
Note that $L_{MC}(A)=0$ if and only if $\card{A}\leq 1$.
Before Choquet, in 1931 Karl Menger introduced this notion for arcs in metric spaces \cite[pp.~741--742]{Menger1931} and asked whether it coincides with arc length.
Shortly after, Yukio Mimura \cite{Mimura1933} gave a proof for arcs in Euclidean space.
Choquet independently introduced $L_{MC}$ in \cite{Choquet1938} for closed subsets of Euclidean space, and announced the following result.
Unfortunately, he did not include a proof.
(See Section~\ref{history} for further historical remarks.)
\begin{choquettheorem}\label{choquet}
For any continuum $A$ in Euclidean space, \[L_{MC}(A)=L^*(A).\]
\end{choquettheorem}
In Theorem~\ref{maintheorem} below we state a more general version of Choquet's Theorem.
We first introduce a related functional, due to Menger.
Note that $L_{MC}(A)$ is not a function of the metric space $A$ on its own, but depends on the metric space $X$ in which $A$ is contained.
We can attempt to fix this by restricting the Steiner points to be in $A$.
Thus, for any finite $P\subseteq X$ and any $A\subseteq X$ define
\settowidth{\myl}{$G=(V,E)$ is a connected graph}
\[\operatorname{smt}_A(P)=\inf\setbuilder{\ell(G)}{\text{$G=(V,E)$ is connected with $P\subseteq V$ and $V\setminus P\subseteq A$}}.\]
As before, we may restrict the graphs in the definition of $\operatorname{smt}_A(P)$ to be Steiner trees on $P$.
Also, $\operatorname{smt}_A(P)\leq\operatorname{smt}_B(Q)$ whenever $P\subseteq Q$ and $A\supseteq B$.
For any $A\subseteq X$, define its \emph{intrinsic Menger length} to be
\[ L_{IM}(A) = \sup\setbuilder{\operatorname{smt}_A(P)}{\text{$P$ is a finite subset of $A$}}.\]
Menger \cite[pp.~741--742]{Menger1931} introduced $L_{IM}$ for arcs in metric spaces.
Clearly, $L_{MC}(A)\leq L_{IM}(A)$ for any $A\subseteq X$.
Like $L_{MC}$, $L_{IM}$ is defined for arbitrary subsets of $X$.
For instance, if $A$ is the vertex set of a square of side length $1$, then $L_{IM}(A)=3$, while if $B=A\cup\set{c}$, where $c$ is the centre of the square, then $L_{IM}(B)=2\sqrt{2}$ (Fig.~\ref{frinkexample}).
This shows that, unlike $L_{MC}$, $L_{IM}$ is not monotone, hence not lower semicontinuous in the sense of Proposition~\ref{MClsc}.
(See Section~\ref{topology} for a detailed discussion of its continuity properties.)
Nevertheless, it follows from Choquet's Theorem that if $A$ is a continuum, then $L_{MC}(A)$ is in fact independent of the ambient space $X$.
The following generalisation of Choquet's Theorem is the main result of this paper.
\begin{theorem}\label{maintheorem}
For any subset $A$ of a metric space $X$, $L^*(A)\leq L_{MC}(A)\leq L_{IM}(A)$.
If $A$ is furthermore connected, then $L^*(A)=L_{MC}(A)=L_{IM}(A)$.
\end{theorem}
The proof, which uses nothing more than elementary properties of graphs and metric spaces, is in the next section.
As a corollary we obtain Theorem~\ref{lsc} by showing that $L_{MC}$ is lower semicontinuous on all subsets of $X$ in Lemma~\ref{steiner2} and Proposition~\ref{MClsc}.
\begin{lemma}\label{steiner2}
For all finite non-empty subsets $P$ and $Q$ of $X$, $\operatorname{smt}(P)\leq\operatorname{smt}(Q)+\card{P}e(P,Q)$.
\end{lemma}
\begin{proof}
Let $T_Q=(V_Q,E_Q)$ be a Steiner tree on $Q$.
For each $p\in P$, let $v(p)$ be a point in $Q$ closest to $p$.
Let $V=P\cup Q$ and $E=E_Q\cup\setbuilder{\set{p,v(p)}}{p\in P\setminus Q}$.
Then $T=(V,E)$ is a Steiner tree on $P$ of length
\[\ell(T) = \ell(T_Q)+\sum_{p\in P}d(p,v(p)) \leq \ell(T_Q)+\card{P}e(P,Q).\]
Thus, $\operatorname{smt}(P)\leq\ell(T_Q)+\card{P}e(P,Q)$ for all Steiner trees $T_Q$ on $Q$.
It follows that $\operatorname{smt}(P)\leq\operatorname{smt}(Q)+\card{P}e(P,Q)$.
\end{proof}
\begin{proposition}\label{MClsc}
Let $A$ be a subset of a metric space $X$.
\begin{enumerate}
\item If $L_{MC}(A)<\infty$ then for each $\varepsilon>0$ there exists a finite collection~$\mathscr{U}$ of open sets in $X$ hit by $A$ such that for any $B\subseteq X$ that hits~$\mathscr{U}$, $L_{MC}(A)<L_{MC}(B)+\varepsilon$.
\item If $L_{MC}(A)=\infty$, then for each $M>0$ there exists a finite collection~$\mathscr{U}$ of open sets in $X$ hit by $A$ such that for any $B\subseteq X$ that hits~$\mathscr{U}$, $L_{MC}(B)>M$.
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose first that $L_{MC}(A)<\infty$.
Let $\varepsilon>0$.
Choose a finite $P\subseteq A$ such that $\operatorname{smt}(P) > L_{MC}(A)-\varepsilon/2$.
Let $\delta=\varepsilon/(2\card{P})$.
Then $A$ trivially hits $\mathscr{U}=\setbuilder{B(p,\delta)}{p\in P}$.
Let $B\subseteq X$ be an arbitrary set that hits $\mathscr{U}$, and for each $p\in P$, choose $p'\in B\cap B(p,\delta)$.
Let $P'=\setbuilder{p'}{p\in P}$.
Then $e(P,P')<\delta$, and by Lemma~\ref{steiner2}, $\operatorname{smt}(P)<\operatorname{smt}(P')+\card{P}\delta\leq L_{MC}(B)+\varepsilon/2$.
It follows that $L_{MC}(A) < L_{MC}(B)+\varepsilon$.

The case where $L_{MC}(A)=\infty$ can be proved by making minimal adjustments to the above proof.
\end{proof}
As another corollary of Theorem~\ref{maintheorem} we obtain a theorem of Bogn\'ar \cite{Bognar1989} (cf.\ Fremlin \cite[4A]{Fremlin1992}).
\begin{corollary}[Bogn\'ar \cite{Bognar1989}]\label{bognartheorem}
For any connected subset $A$ of a metric space $X$, $L^*(A)=L^*(\overline{A})$.
\end{corollary}
This corollary is immediate from Theorem~\ref{bognar} and the fact that the lower limit of a constant sequence $\net{A}$ is the closure $\closure{A}$.
It also follows from Theorem~\ref{maintheorem} by noting that, since $\operatorname{smt}(\cdot)$ is a continuous function on the collection of finite subsets of $X$, $L_{MC}(A)=L_{MC}(\overline{A})$ for any subset $A$ of a metric space $X$.

We note in passing that if $A$ is connected and $L^*(A)<\infty$, then $A$ is $L^*$-measurable \cite[4I]{Fremlin1992}, a result which is beyond the scope of this paper.

We may ask whether the use of Steiner points is necessary.
In fact, before Menger introduced $L_{MC}$ and $L_{IM}$ for arcs in \cite{Menger1931, Menger1932}, he defined \cite[\S\ 6]{Menger1930} the length of a metric continuum $C$ in the following straightforward way:
\[ L_M(C) =\sup\setbuilder{\operatorname{mst}(P)}{\text{$P$ is a finite non-empty subset of $C$}},\]
where \[\operatorname{mst}(P)=\operatorname{smt}_{\emptyset}(P)=\min\setbuilder{\ell(G)}{\text{$G=(P,E)$ is connected}}\] is the length of a minimal spanning tree on $P$.
Menger \cite{Menger1930} showed that if $C$ is an arc, $L_M(C)=L^*(C)$, and also that $L_M$ is lower semicontinuous on the collection of continua.
Unlike $L_{MC}(C)$, $L_M(C)$ depends only on $C$, and not on the metric space in which $C$ lies.
However, as pointed out by Frink \cite{Frink1936}, $L_M(C)$ in general does not necessarily equal $L^*(C)$.
To demonstrate this, Frink used Menger's own observation that $L_M$ is not monotone even if $C$ is a continuum (Fig.~\ref{frinkexample}).
\begin{figure}
\centering
\begin{tikzpicture}[thick, scale=1.5]
    \foreach \x in {-45,45,135,225}
        {\draw[thin,gray] (0,0) -- (\x:1cm);}
    \draw (-45:1cm) \foreach \x in {45,135,225} {-- (\x:1cm)} -- cycle;
    \draw (-0.5,0) node{$1$};
    \draw (0,-1) node{$A$};
    \foreach \x in {-45,45,135,225}
        {\fill (\x:1cm) circle (1.5pt);}
\begin{scope}[xshift=3cm]
    \draw[thin,gray] (-45:1cm) \foreach \x in {45,135,225} {-- (\x:1cm)} -- cycle;
    \foreach \x in {-45,45,135,225}
        {\draw (0,0) -- (\x:1cm);}
    \draw (0,-1) node{$B$};
    \foreach \x in {-45,45,135,225}
        {\fill (\x:1cm) circle (1.5pt);}
    \fill (0,0) circle (1.5pt) node[right=2mm] {$c$};
\end{scope}
\draw(1.5,0) node {$\subset$};
\draw(1.5,-1.5) node {$\operatorname{smt}_A(A)=\operatorname{mst}(A)=3>\operatorname{smt}_B(B)=\operatorname{mst}(B)=2\sqrt{2}$};
\end{tikzpicture}
\caption{Menger's example \cite[top of p.~476]{Menger1930} demonstrating that $P\mapsto \operatorname{mst}(P)$ and $P\mapsto\operatorname{smt}_P(P)$ are not monotone}\label{frinkexample}
\end{figure}
Let $C$ be the union of the two diagonals of a unit square in the plane.
Then $L^*(C)=2\sqrt{2}$, but $L_M(C)=3$, as it is not too difficult to show that for any finite non-empty $P\subseteq C$, $\operatorname{mst}(P)\leq 3$ with equality if and only if $P=A$, the set consisting of the four vertices of the square.
(Note that $\operatorname{smt}(A)=1+\sqrt{3} < 2\sqrt{2}=\operatorname{mst}(B)$, where $B=A\cup\set{c}$, with $c$ the centre of the square.)

Nevertheless, it follows from Lemma~\ref{moore} below that \[L_{MC}(A)\leq L_{IM}(A)\leq L_M(A)\leq 2L_{MC}(A)\] for any $A\subseteq X$, and therefore, if any one of these quantities is finite, all of them are.
There is a simple characterization of metric spaces $A$ for which any of these quantities is finite.
They are exactly the subsets of metric continua of finite outer linear measure.
\begin{theorem}\label{shortest1}
Any metric space $A$ with $L_{MC}(A)<\infty$ can be embedded into a metric continuum $C$ such that $L_{MC}(A)=L_{MC}(C)=L^*(C)$.
\end{theorem}
The proof, although outside the scope of this paper, uses Gromov's compactness theorem for Gromov--Hausdorff convergence \cite{Ducret2008, PaoliniStepanov2013}.
We also mention the following result that can be proved in a standard way with the help of the Blaschke selection theorem for Hausdorff convergence.
As mentioned before, a metric space $X$ is \emph{convex} if any two points $a,b\in X$ are joined by an arc of length $d(a,b)$.
A metric space is called \emph{proper} if all closed and bounded sets are compact.
\begin{theorem}\label{shortest2}
Let $X$ be a convex, proper metric space.
For any $A\subseteq X$, if $L_{MC}(A)<\infty$ then $A$ is contained in a continuum $C\subseteq X$ such that $L_{MC}(A)=L_{MC}(C)=L^*(C)$.
\end{theorem}
In Section~\ref{history} we discuss more general results on the existence of connected sets of minimum length containing a given set.
\section{Proof of Theorem~\ref{maintheorem}}\label{proof}
Recall that a Steiner tree is called proper if the degree of each Steiner point is at least $2$.
As mentioned before, it is very important for our purposes to allow Steiner points of degree $2$.
However, as is well known, in any Steiner tree on a set $P$, there are at most $\card{P}-2$ Steiner points of degree at least $3$.
This follows from the fact that a tree on $\card{V}$ vertices has $\card{V}-1$ edges and degree counting.
A \emph{chain} in a Steiner tree on $P$ is a path in the tree such that all vertices of the path, except possibly the end-vertices, are Steiner points of degree $2$.
A chain in a Steiner tree is called \emph{maximal} if it is not properly contained in a larger chain.
Let $T$ be a proper Steiner tree.
Each vertex of $T$ is contained in some maximal chain of $T$, and each edge of $T$ is contained in a unique maximal chain.
Thus, any two maximal chains are edge-disjoint.
Also, if two maximal chains have a common vertex, this vertex must be a common endpoint that is not a Steiner point of degree $2$.
(See Fig.~\ref{chain}.)
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.5]
\draw (0,0) -- (0.25,0.30) -- (0.5,0.55)  -- (0.78,0.72)  -- (1,1)  -- (1.3,0.87)  -- (1.7,0.9)  -- (2,1)  -- (2.30,0.75)  -- (2.60,0.5)  -- (2.80,0.25)  -- (3,0) ;
\draw (0,2)  -- (0.20,1.75)  -- (0.53,1.5)  -- (0.70,1.25)  -- (1,1);
\draw (2,1) -- (2.25,1.30)  -- (2.5,1.52)  -- (2.75,1.80)  -- (3,2) ;

\fill (0,0) circle (1.5pt);
\fill (0.25,0.30) circle (1pt);
\fill (0.5,0.55) circle (1pt);
\fill (0.78,0.72) circle (1pt);

\fill (1,1) circle (1pt);

\fill (1.3,0.87) circle (1pt);
\fill (1.7,0.9) circle (1pt);

\fill (2,1) circle (1pt);

\fill (2.30,0.75) circle (1pt);
\fill (2.60,0.5) circle (1pt);
\fill (2.80,0.25) circle (1pt);
\fill (3,0) circle (1.5pt);

\fill (0,2) circle (1.5pt);
\fill (0.20,1.75) circle (1pt);
\fill (0.53,1.5) circle (1pt);
\fill (0.70,1.25) circle (1pt);

\fill (2.25,1.30) circle (1pt);
\fill (2.5,1.52) circle (1pt);
\fill (2.75,1.80) circle (1pt);
\fill (3,2) circle (1.5pt);

\draw[thick, blue] (-0.2,0) .. controls (-0.3,0) and (0.3,0.5) .. node[above left] {$C_4$}
(0.8,1) .. controls (0.9,1.1) and (1.1,1.2) ..
(1.15,1) .. controls (1.3,0.9) and (0.7,0.5) ..
(0.2,0) .. controls (0.1,-0.1) and (-0.15,-0.25) .. (-0.2,0) -- cycle;

\draw[thick, blue] (-0.2,2) .. controls (-0.3,2) and (0.3,1.5) .. node[below left] {$C_1$}
(0.8,1) .. controls (0.9,0.9) and (1.1,0.8) ..
(1.15,1) .. controls (1.3,1.1) and (0.7,1.5) ..
(0.2,2) .. controls (0.1,2.1) and (-0.15,2.25) .. (-0.2,2) -- cycle;

\draw[thick, blue] (3.2,0) .. controls (3.3,0) and (2.9,0.5) .. node[above right] {$C_5$}
(2.2,1) .. controls (2.1,1.1) and (1.9,1.2) ..
(1.85,1) .. controls (1.7,0.9) and (2.5,0.5) ..
(2.8,0) .. controls (2.9,-0.1) and (3.15,-0.25) .. (3.2,0) -- cycle;

\draw[thick, blue] (3.2,2) .. controls (3.3,2) and (2.7,1.5) .. node[below right] {$C_3$}
(2.2,1) .. controls (2.1,0.9) and (1.9,0.8) ..
(1.85,1) .. controls (1.7,1.1) and (2.3,1.5) ..
(2.8,2) .. controls (2.9,2.1) and (3.15,2.25) .. (3.2,2) -- cycle;

\draw[thick, blue] (1,0.8) .. controls (0.8,0.8) and (0.8,1.1) ..
(1.05,1.16) .. controls (1.3,0.9) and (1.7,0.95) .. node[above] {$C_2$}
(2,1.16) .. controls (2.2,1.2) and (2.2,0.8) ..
(2,0.8) .. controls (1.7,0.7) and (1.3,0.7) .. (1,0.8) -- cycle;

\end{tikzpicture}
\caption{A proper Steiner tree on a set of $4$ points can be covered by $5$ maximal chains.}\label{chain}
\end{figure}
We can replace each maximal chain in a proper Steiner tree $T$ by a single edge joining the endpoints of the chain to obtain a \emph{reduced} Steiner tree $T'$ on the same set where each Steiner point has degree at least $3$, and such that $\ell(T')\leq \ell(T)$ (by the triangle inequality).
\begin{lemma}\label{chain1}
A proper Steiner tree $T=(V,E)$ on a finite set $P$ has at most $2\card{P}-3$ maximal chains.
\end{lemma}
\begin{proof}
This follows from the above discussion by noting that in a reduced Steiner tree there are at most $\card{P}-2$ Steiner points, hence there are at most $2\card{P}-2$ vertices in total, hence there are at most $2\card{P}-3$ edges.
\end{proof}
The following is a well-known result occurring implicitly or explicitly in \cite{Wazewski1927, Kolmogorov1933, Milgram1940, Eilenberg1943, Gilbert-Pollak1968}.
\begin{lemma}\label{moore2}
Let $P$ be a finite subset of a metric space $X$ with $\card{P}\geq 3$.
Then for any Steiner tree $T$ on $P$, there exists a cycle $C$ with vertex set $P$ such that $\ell(C)\leq 2\ell(T)$.
\end{lemma}
\begin{proof}
By possible removing edges from $T$, we may assume that $T$ is proper.
Also, we may replace $T$ by a reduced Steiner tree without increasing its length.
It is therefore sufficient to show the lemma for reduced Steiner trees on $P$.
We use induction on $\card{P}\geq 3$.
For the case $\card{P}=3$, note that there are two possible reduced Steiner trees on $P$, one with no Steiner points, and one with a single Steiner point joined to each point in $P$.
In both cases, the inequality follows from the triangle inequality.

In the case $\card{P}>3$, if we remove each point in $P$ of degree $1$ from $T$, the remaining graph is still a tree, hence has a vertex $v$ of degree $1$.
This vertex is necessarily a Steiner point, hence is joined to two points $p_1,p_2\in P$.
Then the tree $T'$ obtained from $T$ by removing the points $p_1$, $p_1$, and the edges $\set{p_1,v}$ and $\set{p_2,v}$ is a Steiner tree on $P'=P\cup\set{v}\setminus\set{p_1,p_2}$, and by induction, there is a cycle $C'$ on $P'$ such that $\ell(C')\leq2\ell(T')$.
Let $C$ be the cycle on $P$ obtained by replacing the two edges $\set{v,u}$ and $\set{v,w}$ incident to $v$ by $\set{u,p_1}$, $\set{p_1,p_2}$, $\set{p_2,w}$.
Then it follows from the triangle inequality that $\ell(C)\leq 2\ell(T)$.
\end{proof}
\begin{lemma}\label{moore}
For any finite non-empty $P\subseteq X$, \[\operatorname{mst}(P)\leq \frac{2(\card{P}-1)}{\card{P}}\operatorname{smt}(P).\]
\end{lemma}
\begin{proof}
The lemma is trivial if $\card{P}=1$, and follows from the triangle inequality for $\card{P}=2$.
Let $\varepsilon>0$.
Let $T$ be a Steiner tree on $P$ with $\ell(T) < \operatorname{smt}(P)+\varepsilon$.
By Lemma~\ref{moore2} there exists a cycle through $P$ of length at most $2\ell(T)$.
By removing the longest edge from this cycle we obtain a path through $P$ of length at most $\frac{\card{P}-1}{\card{P}}2\ell(T)$, that is,
\[ \operatorname{mst}(P)\leq \frac{\card{P}-1}{\card{P}}2\ell(T) < \frac{\card{P}-1}{\card{P}}2(\operatorname{smt}(P)+\varepsilon).\]
Since this holds for any $\varepsilon>0$, we obtain the conclusion.
\end{proof}

Let $S\subseteq A\subseteq X$ and $\varepsilon>0$.
Then $S$ is called \emph{$\varepsilon$-separated} if $d(x,y)\geq\varepsilon$ for all distinct $x,y\in S$.
Also, $S$ is called an \emph{$\varepsilon$-net} of $A$ if each point of $A$ has distance $<\varepsilon$ to some point of $S$.
Note that a maximal $\varepsilon$-separated subset of $A$ is also an $\varepsilon$-net of $A$.
The set $A$ is said to be \emph{totally bounded} if $A$ contains a finite $\varepsilon$-net for each $\varepsilon>0$.
\begin{lemma}\label{totallybounded}
Let $A\subseteq X$ satisfy $L_{MC}(A)<\infty$.
Then for any $\varepsilon>0$, any $\varepsilon$-separated subset of $A$ has at most $\max\set{\frac{2}{\varepsilon}L_{MC}(A),1}$ points.
In particular, $A$ is totally bounded.
\end{lemma}
\begin{proof}
Let $P$ be a finite $\varepsilon$-separated subset of $A$, that is, $d(x,y)\geq\varepsilon$ for all distinct $x,y\in P$.
Without loss of generality, $\card{P}\geq 2$.
Then $\operatorname{mst}(P)\geq (\card{P}-1)\varepsilon$, since a spanning tree of $P$ has $\card{P}-1$ edges, and by the previous lemma,
\[\operatorname{smt}(P) \geq \frac{\card{P}}{2(\card{P}-1)}\operatorname{mst}(P)
\geq \frac{\card{P}}{2(\card{P}-1)}(\card{P}-1)\varepsilon = \frac{\varepsilon}{2}\card{P}.\]
It follows that \[\card{P}\leq \frac{2}{\varepsilon}\operatorname{smt}(P)\leq\frac{2}{\varepsilon}L_{MC}(A).\]
Therefore, if we choose points of $A$ one by one, keeping the chosen subset $\varepsilon$-separated, we will stop after at most $\max\set{1,\frac{2}{\varepsilon}L_{MC}(A)}$ steps with a maximal $\varepsilon$-separated subset of $A$.
\end{proof}

\begin{lemma}\label{chain2}
Consider a chain $C=x_1x_2\dots x_n$ of length $\ell(C)$ in a Steiner tree.
Then for any $t>0$ there exist $k < 1+2\ell(C)/t$ vertex-disjoint subchains $C_1,\dots,C_k$ such that each $\ell(C_i)\leq t$, the vertex set of $C$ is partitioned by the vertex sets of the $C_i$ and all edges of $C$ not in any $C_i$ have length $>t$.
\end{lemma}
\begin{proof}
We use the following simple online bin-packing algorithm.
Walk along $C$ from $x_1$ to $x_n$, cutting it up at vertices into chains that are as long as possible, trying to keep their lengths $\leq t$.
Whenever an edge of length $>t$ is found, it is taken to be a single ``outsize'' chain (Fig.~\ref{fig:chain}).
\begin{figure}
\centering
\begin{tikzpicture}[thick,scale=0.9]
\draw[|-|] (0,1) -- node[midway,fill=white] {$t$} (3,1);
\draw[|-|] (2.3,-0.5) -- node[midway,fill=white] {$t$} (5.3,-0.5);
\draw[|-|] (6,-0.5) -- node[midway,fill=white] {$t$} (9,-0.5);
\draw[very thick] (0,0) -- (9.3,0) node [right] {\dots};
\fill (0,0) circle (2pt);
\fill (1.5,0) circle (2pt);
\fill (2.3,0) circle (2pt);
\draw[latex-latex] (0,0.5) -- node[midway,fill=white] {$C_1$} (2.3,0.5);
\fill (6,0) circle (2pt);
\fill (6.5,0) circle (2pt);
\fill (7,0) circle (2pt);
\fill (7.6,0) circle (2pt);
\fill (8.3,0) circle (2pt);
\draw[latex-latex] (2.3,0.5) -- node[midway,fill=white] {$C_2$} (6,0.5);
\draw[latex-latex] (6,0.5) -- node[midway,fill=white] {$C_3$} (8.3,0.5);
\fill (9.3,0) circle (2pt);

\end{tikzpicture}
\caption{Online chain cutting}\label{fig:chain}
\end{figure}
Denote the consecutive chains made in this way by $C_1,\dots,C_k$.
Because of maximality, the sum of the lengths of any two adjacent chains is $>t$.
It follows that $2\ell(C)>(k-1)t$.
Finally, remove all outsize chains and add a new chain $x_i$ of length $0$ between any two adjacent outsize chains $x_{i-1}x_i$ and $x_ix_{i+1}$.
In this way we remove at least as many chains as we add new chains of length $0$.
We end up with at most $k$ chains, each of length $\leq t$, with vertices covering the vertices of $C$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{maintheorem}]
We first show that $L^*(A)\leq L_{MC}(A)$.
Without loss of generality, $L_{MC}(A)<\infty$.
Then, by Lemma~\ref{totallybounded}, $A$ is totally bounded.
The idea of the proof is as follows.
We take a finite $P\subseteq A$ with $\operatorname{smt}(P)$ very close to $L_{MC}(A)$.
Then we extend $P$ to an $\varepsilon$-net $P'$ with $\varepsilon$ very small, depending on $\delta$ and another auxiliary large constant $R$.
Then $\operatorname{smt}(P')$ is still very close to $L_{MC}(A)$.
We take a proper Steiner tree $T'$ on $P'$ of length very close to $\operatorname{smt}(P')$, and use Lemmas~\ref{chain1} and \ref{chain2} to cut the proper subtree $T$ of $T'$ joining $P$ into small pieces to create a good $\delta$-cover of $A$.

Let $\delta>0$ be arbitrary.
Without loss of generality, we may assume $\delta<1/8$.
Fix a finite $P\subseteq A$ with $\operatorname{smt}(P)>L_{MC}(A)-\delta/4$.
Set
\[ \varepsilon = \min \set{ \min_{x,y\in P,x\neq y} d(x,y), \delta^2,\frac{\delta}{\card{P}} }. \]
Then $P$ is $\varepsilon$-separated, and by Lemma~\ref{totallybounded}, $P$ is contained in a maximal $\varepsilon$-separated finite $P'\subseteq A$ which is also an $\varepsilon$-net.
Let $T'=(V',E')$ be a proper Steiner tree on $P'$ such that
\[ \ell(T') < \operatorname{smt}(P') +\frac{\delta}{4}\leq L_{MC}(A)+\frac{\delta}{4}.\]
We decompose $T'$ as follows.
Let $T=(V,E)$ be the union of all the paths in $T'$ between pairs of points from $P$.
Then $T$ is clearly a Steiner tree on $P$.
Remove the edges of $T$ from $T'$ (but not the vertices)
to obtain $G=(V',E'\setminus E)$.
Let $T_v$ be the connected component of $G$ that contains $v\in V$.
Denote the maximal chains of $T$ by $C_i$, $i=1,\dots,c$ (see Fig.~\ref{decomposition}).
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.9]
\draw (1,1) -- (0.7,1.1) -- (0.5,1.4);
\draw (0.7,1.1) -- (0.4,1);
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\draw (3,0) -- (3.3,-0.2);
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\draw[thick, blue] (3.2,0) .. controls (3.3,0) and (2.9,0.5) .. node[above right] {$C_3$}
(2.2,1) .. controls (2.1,1.1) and (1.9,1.2) ..
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\draw[thick, blue] (3.2,2) .. controls (3.3,2) and (2.7,1.5) .. node[below right] {$C_2$}
(2.2,1) .. controls (2.1,0.9) and (1.9,0.8) ..
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(2,1.16) .. controls (2.2,1.2) and (2.2,0.8) ..
(2,0.8) .. controls (1.7,0.7) and (1.3,0.6) .. (1,0.7) -- cycle;

\draw[thick, red] (0.7,1.1) circle (5mm); \draw[red] (0.1,0.9) node {$T_a$};
\draw[thick, red] (3.1,2.2) ellipse (8mm and 3.5mm); \draw[red] (2.3,1.9) node {$T_b$};
\draw[thick, red] (3.1,-0.15) ellipse (5mm and 3mm); \draw[red] (2.4,-0.2) node {$T_c$};

\end{tikzpicture}
\caption{Decomposing the Steiner tree $T'$ with $P=\set{a,b,c}$}\label{decomposition}
\end{figure}
By Lemma~\ref{chain1}, $c\leq2\card{P}-3$.
We use Lemma~\ref{chain2} to cut each chain $C_i$ up into disjoint subchains $C_{i,1},\dots,C_{i,k(i)}$, each of length $\ell(C_{i,j})\leq \delta/2$, where the number of pieces for each $C_i$ is $k(i)< 1+\frac{4\ell(C_i)}{\delta}$.
Then the $C_{i,j}$ and the $T_v$ together cover the vertex set of $T'$, which includes the $\varepsilon$-net $P'$.

Some of the $T_v$ are so small that they may be ignored, as follows.
We say that $T_v$ is \emph{large} if $\ell(T_v)\geq\varepsilon$.
Let $L=\setbuilder{v\in V}{\ell(T_v)\geq\varepsilon}$ be the set of vertices $v$ with a large $T_v$.
Let $a\in A$.
Then there exists $p\in P'\subseteq V'$ with $d(a,p)<\varepsilon$.
If $p$ is not in any chain $C_i$ or large $T_v$, then $p$ is contained in some $T_v$ of length $\ell(T_v)<\varepsilon$.
However, then $d(p,v)<\varepsilon$, hence $d(a,v)<2\varepsilon$ for some $v\in V$.
Therefore, \[U_{i,j}=\bigcup_{v\in V(C_{i,j})} B(v,2\varepsilon)\qquad (1\leq i\leq c, 1\leq j\leq k(i))\]
and \[U_v=\bigcup_{w\in V(T_v)} B(w,\varepsilon)\qquad(v\in L)\]
together cover $A$.
We next estimate the diameters of the $U_{i,j}$ and $U_v$ in this cover.
First,
\[\diam(U_{i,j}) \leq \diam(C_{i,j})+4\varepsilon\leq \ell(C_{i,j})+4\varepsilon\leq\frac{\delta}{2}+4\delta^2 <\delta.\]
Next, the total length of the large $T_v$ is
\begin{align*}
\sum_{v\in L}\ell(T_v) &\leq \ell(T')-\ell(T)\leq\ell(T')-\operatorname{smt}(P)\\ &< \left(L_{MC}(A)+\frac{\delta}{4}\right) - \left(L_{MC}(A)-\frac{\delta}{4}\right)= \frac{\delta}{2}.
\end{align*}
In particular, since $\card{L}\varepsilon\leq\sum_{v\in L}\ell(T_v)$, the number of large $T_v$ is $\card{L} < \delta/(2\varepsilon)$.
Since each $T_v$ is connected, $U_v$ has diameter
\begin{align}
\diam(U_v)&\leq\ell(T_v)+2\varepsilon < \frac{\delta}{2}+2\delta^2 < \delta. \label{B}
\end{align}
It follows that
\[ \setbuilder{U_{i,j}}{1\leq i\leq c, 1\leq j\leq k(i)}\cup\setbuilder{U_v}{v\in L}\]
is a $\delta$-cover of $A$ with
\begin{align*}
&\phantom{{}\mathrel{<}{}} \sum_{i=1}^c\sum_{j=1}^{k(i)}\diam(U_{i,j})+\sum_{v\in L}\diam(U_v)\\
&< \sum_{i=1}^c\sum_{j=1}^{k(i)}(\ell(C_{i,j})+4\varepsilon)+\sum_{v\in L}(\ell(T_v)+2\varepsilon)\\
&< \ell(T)+4\varepsilon\sum_{i=1}^c\left(1+\frac{4\ell(C_i)}{\delta}\right)+2\varepsilon\card{L}\\
&\leq \left(1+\frac{16\varepsilon}{\delta}\right)\ell(T) + 4 \varepsilon c +2\varepsilon\card{L}\\
&\leq \left(1+16\delta\right)\ell(T') + 4 (2\card{P}-3)\varepsilon +\delta\\
&< \left(1+16\delta\right)\left(L_{MC}(A)+\frac{\delta}{4}\right)+8\delta+\delta.
\end{align*}
Therefore, $L^*_\delta(A)\leq \left(1+16\delta\right)\left(L_{MC}(A)+\frac{\delta}{4}\right)+9\delta$.
Letting $\delta\to0$, we obtain \[L^*(A)=\lim_{\delta\to 0} L^*_\delta(A)\leq L_{MC}(A).\]

The second inequality $L_{MC}(A)\leq L_{IM}(A)$ is obvious.

\smallskip
It remains to show that if $A$ is a connected subset of $X$, then $L_{IM}(A)\leq L^*(A)$.
We may assume without loss of generality that $L^*(A)<\infty$.
As already remarked, since neither $L_{IM}$ nor $L^*$ depends on $X$, we may also assume that $A$ is the whole metric space.
Let $\varepsilon>0$ and let $P$ be an arbitrary non-empty finite subset of $A$.
We want to use a good $\delta$-covering of $A$ (for sufficiently small $\delta>0$) to construct a good Steiner tree of $P$ with Steiner points in $A$, in order to show that $\operatorname{smt}_A(P)\leq L^*(A)+\varepsilon$.
Without loss of generality, $\card{P}\geq 2$.
Since $A$ then has at least two points, its connectedness implies that any non-empty open subset of $A$ is infinite.
Let \[\delta=\min\set{\frac{\varepsilon}{2\card{P}}, \min_{x,y\in P, x\neq y}d(x,y)}.\]
Let $\setbuilder{U_i}{i\in\numbersystem{N}}$ be a $\delta$-cover of $A$ such that
\[ \sum_{i=1}^\infty\diam(U_i) < L^*(A)+\frac{\varepsilon}{2}.\]
If we instead take a $(\delta/2)$-cover and replace each $U_i$ by $\bigcup_{x\in U_i}B(x,\eta/2^i)$ for a sufficiently small $\eta>0$, we may assume that each $U_i$ is open in $A$.
By the choice of $\delta$, $\card{U_i\cap P}\leq 1$ for all $i\in\numbersystem{N}$.
We say that two points $x,y\in A$ are \emph{joined by $\setbuilder{U_i}{i\in\numbersystem{N}}$} if there exists a finite sequence $U_{i(1)},\dots,U_{i(k)}$ such that $x\in U_{i(1)}$, $y\in U_{i(k)}$, and for each $t=1,\dots,k-1$, $U_{i(t)}\cap U_{i(t+1)}\neq\emptyset$.
The connectedness of $A$ implies that any two $x,y\in A$ are connected by $\set{U_i}$.
Indeed, for a fixed $x\in A$, the set $S$ of all $y\in A$ such that $x$ and $y$ are connected by $\set{U_i}$ is open, as it is a union of $U_i$.
The complement $A\setminus S$ is also open, since if $z\in A\setminus S$, then $z\in U_j$ for some $j$ and then $U_j\subseteq A\setminus S$, otherwise there would exist $y\in U_j\cap S$, hence the sequence $U_{i(1)},\dots,U_{i(k)}$ demonstrating that $x$ and $y$ are joined, together with $U_{i(k+1)}=U_j$, show that $z\in S$, a contradiction.

For any $x,y\in A$, choose a sequence $U_{i(1,x,y)},\dots,U_{i(k(x,y),x,y)}$ demonstrating that $x$ and $y$ are joined, where we choose a single $U_{i(1,x,x)}$ when $x=y$.
Then the graph $\mathscr{G}$ on the set $\mathscr{V}=\setbuilder{i(j,x,y)}{x,y\in A, 1\leq j\leq k(x,y)}$ with edges $\mathscr{E}=\setbuilder{\set{i,j}}{U_i\cap U_j\neq\emptyset}$ is connected, hence contains a minimal subgraph that contains $\setbuilder{i(1,x,x)}{x\in A}$.
This subgraph is necessarily a tree $\mathscr{T}=(I,\mathscr{E}')$, with $I$ a non-empty finite subset of $\numbersystem{N}$ and $\mathscr{E}'$ a finite subset of $\mathscr{E}$.
For each edge $e=\set{i,j}\in\mathscr{E}'$, choose $x_{e}\in (U_i\cap U_j)\setminus P$ (recall that $U_i\cap U_j$ is infinite).
For each $i\in I$, let $V_i=(P\cap U_i)\cup\setbuilder{x_{e}}{i\in e\in\mathscr{E}'}$.
Then
\begin{equation}\label{star}
\card{V_i} \leq \begin{cases}
\deg_\mathscr{T}(i) &\text{if $P\cap U_i=\emptyset$,}\\
\deg_\mathscr{T}(i)+1 &\text{if $P\cap U_i\neq\emptyset$.}
\end{cases}
\end{equation}
Let $i\in I$.
Since $\deg_\mathscr{T}(i)\geq 1$, $\card{V_i}\geq 1$.
If $\card{V_i}=1$, then $\deg_\mathscr{T}(i)=1$ and $P\cap U_i=\emptyset$.
However, then $i$ and the edge of $\mathscr{T}$ incident with $i$ are redundant, contradicting the minimality of $\mathscr{T}$.
Therefore, $\card{V_i}\geq 2$ for each $i\in I$.
Form a graph $G$ on the vertex set $V=\bigcup_{i\in I}V_i$ by joining the vertices in each $V_i$ by an arbitrary path.
Since $\mathscr{T}$ is connected, $G$ is also connected.
Note that two $V_i$ could intersect in more than one point, hence $G$ is not necessarily a tree.
Nevertheless, any spanning tree $T$ of $G$ will be a Steiner tree on $P$, and we obtain
\begin{align*}
\ell(T)\leq \ell(G) &\leq \sum_{i\in I}(\card{V_i}-1)\diam(U_i)\\
&= \sum_{i\in I}\diam(U_i) + \sum_{i\in I}(\card{V_i}-2)\diam(U_i)\\
&\leq L^*(A)+\frac{\varepsilon}{2} + \delta\sum_{i\in I}(\card{V_i}-2).
\end{align*}
By \eqref{star} and $\card{\mathscr{E}'}=\card{I}-1$,
\begin{equation*}
\sum_{i\in I}(\card{V_i}-2) \leq \sum_{i\in I}(\deg_\mathscr{T}(i)-2)+\card{P}
= -2+\card{P}.
\end{equation*}
It follows that
\begin{equation*}
\operatorname{smt}_A(P) \leq \ell(T) \leq L^*(A) +\frac{\varepsilon}{2} + (\card{P}-2)\delta
\leq L^*(A) +\varepsilon.
\end{equation*}
This holds for arbitrary $P\subseteq A$ and $\varepsilon>0$, and we conclude that $L_{IM}(A)\leq L^*(A)$.
\end{proof}
\section{Lower semicontinuous with respect to which topology?}\label{topology}
Given a topological space $(\mathscr{X},\mathscr{T})$, a function $f\colon \mathscr{X}\to\numbersystem{R}$ is defined to be \emph{lower semicontinuous} at a point $x_0\in \mathscr{X}$ if for any $\varepsilon>0$ there exists an open neighbourhood $U$ of $x_0$ such that for all $x\in U$, $f(x_0)<f(x)+\varepsilon$.
Go\l\k{a}b's original theorem and each of the Theorems~\ref{golab}, \ref{frink}, \ref{bognar}, \ref{fremlin}, \ref{lsc} assert the lower semicontinuity of $L^*$ on the collection of connected subsets of a metric space for an appropriate topology on this collection.
For instance, the well-known \emph{Hausdorff topology} is the topology on the collection $\mathscr{C}(X)$ of all closed subsets of $X$ generated by the basis of all sets of the form
\[ \mathscr{B}(A,r) = \setbuilder{S\in\mathscr{C}(X)}{D(A,S)<r}, \qquad A\in\mathscr{C}(X), r>0.\]
Go\l\k{a}b's theorem states that on the subspace of all continua of $X$, $L^*$ is lower semicontinuous with respect to the Hausdorff topology.
Theorem~\ref{golab} can also be stated as the lower semicontinuity of $L^*$, but now on the coarser \emph{lower Hausdorff topology}, which we define on the space $\mathscr{P}(X)$ of all subsets of $X$.
This is the topology $\mathscr{H}^-$ generated by the basis consisting of all sets of the form
\[ \mathscr{B}^-(A,r) = \setbuilder{S\in\mathscr{P}(X)}{e(A,S)<r}, \qquad A\in\mathscr{P}(X), r>0.\]
Then Theorems~\ref{golab} and \ref{frink} assert the lower semicontinuity of $L^*$ on the subspace of $(\mathscr{P}(X),\mathscr{H}^-)$ consisting of the connected subsets of $X$.

For Theorem~\ref{bognar}, \ref{fremlin} and \ref{lsc}, we introduce the
\emph{lower Vietoris topology}.
This is the topology $\mathscr{V}^-$ on $\mathscr{P}(X)$ generated by the subbasis consisting of all sets of the form
\[ A^- = \setbuilder{S\in\mathscr{P}(X)}{S\cap A\neq\emptyset}, \qquad A\text{ open in }X.\]
Theorems~\ref{bognar}, \ref{fremlin} and \ref{lsc}  assert the lower semicontinuity of $L^*$ on the subspace of $(\mathscr{P}(X),\mathscr{V}^-)$ consisting of the connected subsets of $X$.
Since the lower Vietoris topology is coarser than the lower Hausdorff topology \cite[Lemma~3.2]{Michael1951} (see also \cite[Prop.~4.2.1(ii)]{Klein1984}), Theorems~\ref{bognar}, \ref{fremlin} and \ref{lsc} are formally stronger than Theorems~\ref{golab} and \ref{frink}.
However, the difference is slight, as these two topologies coincide if and only if the metric space $X$ is totally bounded \cite[Theorem~5]{Levi1993}, and for general metric spaces $X$ the two topologies coincide on the subset of $\mathscr{P}(X)$ of all totally bounded subsets of $X$.
(This can be proved analogously to Prop.~4.2.2(ii) of \cite{Klein1984}.)
Since sets that are not totally bounded have infinite outer linear measure, it follows that at points of $\mathscr{P}(X)$ that have finite outer linear measure, the two topologies are the same.
Therefore, the first part of Theorem~\ref{lsc} has the same strength as the first part of Theorem~\ref{frink}.

\section{Historical remarks}\label{history}
\subsection{The length of continua}
The definition of outer linear measure was originally introduced by Carath\'eodory \cite{Caratheodory1914} and generalised by Hausdorff \cite{Hausdorff1918} to arbitrary (including fractional) dimensions.

Menger \cite{Menger1931, Menger1932}, Mimura \cite{Mimura1933} and Choquet \cite{Choquet1938} were early pioneers in the study of Steiner trees (see the historical surveys \cite{Schrijver2005} and \cite{BGTZ}).
No doubt unaware of Choquet's paper, Leonard Blumenthal included Menger's question whether $L_{MC}$ equals arc length in general metric spaces in his book on distance geometry \cite[pp.~66--67]{BlumenthalBook}.
His student, William Ettling \cite{Ettling1978}, claimed to have a proof for all proper metric spaces, but unfortunately his proof, which follows Mimura's first proof very closely,
made the tacit assumption that $X$ is \emph{convex} in the sense that any two points $x,y\in X$ are joined by an arc of length $d(x,y)$.
It is also not difficult to modify Mimura's first proof to work for any connected subset of any metric space, without assuming compactness of the set or convexity of the space, or having to use Go\l\k{a}b's Theorem.
(Unfortunately, Mimura's second proof uses the lower semicontinuity of the length of continua.)

\subsection{Existence of connected sets of minimum length}
For further existence results along the lines of Theorems~\ref{shortest1} and \ref{shortest2}, see Chapter~4 of the book of Ambrosio and Tilli \cite{Ambrosio2004}, and the papers of Ducret and Troyanov \cite{Ducret2008},
Paolini and Ulivi \cite{PaoliniUlivi2010} and
Paolini and Stepanov \cite{PaoliniStepanov2013}.
Ivanov, Nikonov and Tuzhilin \cite{Ivanov2005} characterize the countable metric spaces that have an infinite spanning tree of finite total length.
Starting with Jones \cite{Jones1990}, a different approach has been taken to characterize the subsets of Euclidean spaces that are contained in rectifiable curves.
For an overview of this approach, see Schul's survey \cite{Schul2007}.

\subsection{Go\l\k{a}b's Theorem and its proofs}
Go\l\k{a}b's own proof \cite{Golab1928} of his theorem depends on various results of Wa\.zewski \cite{Wazewski1927}, one of them stating that a continuum $C$ with $L^*(C)<\infty$ in $\numbersystem{R}^n$ is the image of an absolutely continuous mapping $f\colon[0,L^*(C)]\to\numbersystem{R}^n$ that is non-expansive in each coordinate.
This is related to Lemma~\ref{moore2} (see also \cite[Theorem~2]{Eilenberg1943} and \cite[Theorem~4.4]{Alberti2017}).
His proof furthermore uses compactness in the form of the Arzela--Ascoli Theorem.
As can be seen from the proof in this paper, as well as Frink's proof \cite{Frink1936}, it is not necessary to use compactness or to take subsequences.
Among all proofs of Go\l\k{a}b's Theorem known to us, Frink's proof seems to be the simplest and shortest, and proceeds directly from the definition of outer linear measure.

At a point where Go\l\k{a}b's Theorem would be needed, Besicovitch \cite[Proof of Theorem~12]{Besicovitch1938} merely states that it is ``easy to see\dots''.

Faber, Mycielski and Pedersen \cite{FMP1984} use Go\l\k{a}b's Theorem to show that, given a compact subset $S$ of the plane, there is a shortest connected closed subset of the plane which intersects all lines intersecting $S$.
They give a proof of Go\l\k{a}b's Theorem that depends on another result of Wa\.zewski, namely that a continuum with finite outer linear measure is path-connected.
This is the proof that is presented in Falconer's book \cite[Chapter~3]{Falconer1986}.
Like Go\l\k{a}b's original proof, it uses compactness, but in the form of the Blaschke selection theorem.

Bogn\'ar's proof of his version of Go\l\k{a}b's Theorem \cite{Bognar1989} is somewhat involved and even invokes Zorn's Lemma at some point.
It seems that Bogn\'ar's main purpose was to prove what we have stated as Corollary~\ref{bognartheorem}.
Fremlin \cite{Fremlin1992} gives his own short proof of this corollary and refers to it as \emph{M.~Bogn\'ar's theorem}, although it would be very surprising if it hadn't been known earlier.
Nevertheless, it frequently occurs as a technical result in the literature on existence results in the calculus of variations, for instance, Proposition~2.5 in \cite{DalmasoToader2002} and Lemma~2.6 in \cite{PaoliniStepanov2013}.
According to Go\l\k{a}b \cite{Golab1928}, Wa\.zewski proved the following weaker statement:
\begin{theorem}
Let $C_1\subseteq C_2\subseteq C_3\subseteq $ be an increasing sequence of continua such that $\sup_i L^*(C_i)<\infty$.
Then $L^*(\bigcup_i C_i) = L^*(\overline{\bigcup_i C_i})$.
\end{theorem}
Go\l\k{a}b deduces this statement as an immediate corollary to his theorem.

Ambrosio and Tilli \cite{Ambrosio2004} give a highly analytical proof of Go\l\k{a}b's Theorem, in particular using densities and weak-star convergence of Borel measures.
This proof was corrected by Paolini and Stepanov \cite{PaoliniStepanov2013}.
Alberti and Ottolini \cite{Alberti2017} give a simpler proof which still uses analysis and measure theory, though.

\subsection{Generalizations of Go\l\k{a}b's Theorem}
Ambrosio and Tilli's proof of Go\l\k{a}b's Theorem \cite{Ambrosio2004} is adapted by Paolini and Stepanov \cite{PaoliniStepanov2013} to show the following generalisation of Go\l\k{a}b's Theorem, of which the Euclidean case appears in Dal Maso and Toader \cite{DalmasoToader2002}.
\begin{theorem}
Let $C_n$ be a sequence of closed connected subsets of a complete metric space $X$ and $C$ a closed subset of $X$ such that $\lim_{n\to\infty}H(C_n,C)=0$.
Let $K_n$ be a sequence of closed subsets of $X$ and $K$ a closed subset of $X$ such that $\lim_{n\to\infty}H(K_n,K)=0$.
Then
\[ L^*(C\setminus K) \leq\liminf_n L^*(C_n\setminus K_n).\]
\end{theorem}
A further generalisation is found in Giacomini \cite{Giacomini2002}:
\begin{theorem}
Let $m\in\numbersystem{N}$, $\Omega$ an open and bounded subset of $\numbersystem{R}^2$, and let $\varphi\colon\overline{\Omega}\times\numbersystem{R}^2\to[0,\infty)$ be continuous and such that for each $x\in\overline{\Omega}$, $\varphi(x,\cdot)$ is a norm on $\numbersystem{R}^2$.
Then the functional $K\mapsto\int_K\varphi(x,\nu_x)\,\mathrm{d}H^1(x)$ is lower semicontinuous on the collection of compact subsets of $\overline{\Omega}$ of finite $1$-dimensional Hausdorff measure with at most $m$ connected components.
\end{theorem}
Here $\nu_x$ denotes the unit normal vector of $K$ at $x$ (which exists $H^1$-a.e.\ if $H^1(K)<\infty$).
Giacomini considers $K$ to be a fracture and the above functional is its surface energy.
However, this functional can be interpreted as the ordinary $1$-dimensional Hausdorff measure in the Finsler space on $\overline{\Omega}$ determined by $\varphi$, and therefore, this theorem follows from Go\l{\k{a}}b's Theorem for sets with at most $m$ connected components \cite[Corollary~3.3]{DalmasoToader2002}.

It is easily observed that some sort of connectedness is necessary in Go\l\k{a}b's Theorem.
For example, let $A_n=\set{\frac{1}{n},\frac{2}{n},\dots,\frac{n}{n}}$.
Then $A_n$ converges to $A=[0,1]$, but $L^*(A_n)=0$ and $L^*(A)=1$.

Go\l\k{a}b already notes that there is no straightforward generalisation of his theorem to area.
However, Vitushkin \cite{Vituskin1966} generalises Go\l\k{a}b's Theorem to $k$-dimensional Hausdorff measure in $d$-dimensional Euclidean spaces, by assuming the additional hypothesis that the expected number of connected components of the intersection of $A_n$ by a flat of codimension $k-1$ (with respect to the Haar probability measure on the Grassmannian) is bounded uniformly over $n$.

Dal Maso, Morel and Solimini \cite{Dalmaso1992} generalise Go\l\k{a}b's Theorem for linear measure by assuming that the sets in the sequences uniformly satisfy a certain concentration property.
This is generalised to $k$-dimensional measures by Morel and Solimini \cite[Chapter~10]{Morel1995} and Lops \cite{Lops2005}.

{\small

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So we're back where we started. And by we I mean me. And by me I mean I. I'M BACK TO WHERE I STARTED.
I'm sorry if you're reading this before bed, no one should have to look at that.
I've actually been reading books and articles regarding willpower and habit forming, as well as procrastination, goal setting, and habits of successful people for the past month or so. I was even able to use reading about procrastination to put off starting all of this stuff last month. What can I say, I'm a natural.
I read The Power of Habit, and Solving the Procrastination Puzzle.
I read some articles online about morning routines, the finite nature of willpower, routine changing, understanding roadblocks/self-obstacles/internal objections, and other self-helpy crap.
I also like infographics because they are more fun to look at than a bunch of stupid words.
It means that even though most of us know the things in that infographic (and the things that I've read) are beneficial, a lot of us don't do them. I can't tell you the last time I woke up for anything other than to rush to get ready for work or another commitment. I love smacking snooze like my alarm just cut me off in traffic. A big problem for me is I love to stay up late on the interweb (blackpeoplemeet.com), or watching Netflix (I'm currently binging nightly on Parks and Rec, hence the Andy Dwyer reference). So my number one challenge that I need to meet is waking up earlier.
In order to wake up earlier, I need to go to bed earlier. In order to go to bed earlier, I need to exercise so that I'm tired. In order to exercise enough to make me tired, I need to eat healthier foods to give me energy to fuel my exercise. In order to eat healthier foods, I need to wake up early to prepare and eat a healthy breakfast.
*Also known is being a normal, healthy person, something I don't think I've really ever been. While we all know I'll never be normal, I could probably be healthy.
Here's my idea. I'll just imitate a healthy person until I am a healthy person. That's totally going to work. So what would a healthy person do everyday?
I'll do this until January 8th and then I'll probably switch it up. 31 days sounds like a good start to the rest of my life, right? Thoughts? Questions? New skill idea? Let me know!


Note: This scenario can only be accomplished with the Web Deploy command line.
The -source switch in Web Deploy for migrating a web site is appHostConfig and this switch includes IIS settings such as bindings, certificates, and other top level site settings.
For these examples, the Source server IP is 10.0.0.11 and the Target server is 10.0.0.12.
As I am working with customers in either the test or production environments, I always stress the importance of taking backups on all servers BEFORE changes to recreate a restore point in case issues arise during the procedure. In addition, I use the -whatif statement heavily to fix issues before a migration occurs.
The target server must have the Web Deploy service installed and running in order to communicate from the source server. If the service is not installed, run the Web Deploy installer again and select the Service option.
There are two options presented to migrate the web site. The first option generates a Web Deploy package in the .zip format and this package is copied to the destination server. The second option performs the migration directly from the command prompt and is the fastest method. With both methods, please verify the results from the migration is the expected result by using the –whatif command to ensure a successfully migration.
Note: The command switch -source:appHostConfig is the equivalent to the IIS web site level. This means, include everything in the web site when creating the package.
Inspect the output in the text file by searching on Error and Warning and resolve the issues before generating the package. Again, the pipe command to the file is optional and output from the WhatIf command can be viewed within the command window.
Note: If the website on the Target does not exist, the output from the -whatif statement will show all adds and no deletes.
Inspect the output in the text file by searching on Error and Warning and resolve the issues before generating the package.
This method generates a server to server transfer of entire website settings and files.
Inspect the output in the text file by searching on Error and Warning and resolve the issues before generating the package. The output file is the history of the migration and can be archived.


I love florals Shenae, I'm sure your floral maxi is incredible. I love them chunky heels too. Hugs, Terri xo.
Hi Shenae, I love the pink floral prints ! I am sure you look incredible in your Maxi dress . The chunky heels are so cute, love the fuschia! I am feminine myself so I should start floating! Have a great weekend, Hugs , Terri.


text:
Q: 3D numpy array MinMax Normalization I'd like to MinMax normalize the following 3D numpy array "at 2D-layer level" :
np.array([[[0, 1, 2],
           [3, 4, 5],
           [6, 7, 8]],
          [[0, 1, 2],
           [3, 4, 5],
           [6, 7, 10]],
          [[0, 1, 2],
           [3, 4, 5],
           [6, 7, 12]]])

to obtain :
np.array([[[0. , 0.1, 0.2],
           [0.3, 0.4, 0.5],
           [0.6, 0.7, 1. ]],
          [[0. , 0.1, 0.2],
           [0.3, 0.4, 0.5],
           [0.6, 0.7, 1. ]],
          [[0.        , 0.08333333, 0.16666667],
           [0.25      , 0.33333333, 0.41666667],
           [0.5       , 0.58333333, 1.        ]]])

any idea how if could be done without using loops ? Many thanks in advance !

A: One approach is to use .max as follows:
res = arr / arr.max(axis=(1, 2), keepdims=True)
print(res)

Output
[[[0.125      0.125      0.25      ]
  [0.375      0.5        0.625     ]
  [0.75       0.875      1.        ]]

 [[0.         0.1        0.2       ]
  [0.3        0.4        0.5       ]
  [0.6        0.7        1.        ]]

 [[0.         0.08333333 0.16666667]
  [0.25       0.33333333 0.41666667]
  [0.5        0.58333333 1.        ]]]

A: If you define your array as:
x = np.array([[[0, 1, 2],
           [3, 4, 5],
           [6, 7, 8]],
          [[0, 1, 2],
           [3, 4, 5],
           [6, 7, 10]],
          [[0, 1, 2],
           [3, 4, 5],
           [6, 7, 12]]])

You can use reshape to flatten the array:
(
    x.reshape(x.shape[-1],x.shape[0]*x.shape[1]).T /
    np.max(x.reshape(x.shape[2], x.shape[0]*x.shape[1]), axis=1)
).T.reshape(x.shape)

Here the array is flatten to a 2D array where one can take the max of axis=1.

redpajama_set_name: RedPajamaStackExchange


text:
Xiaomi makes 35th adjustment to reduce burden on international business
Yusuf Balogun
Lei Jun on product Launch | Image credit: ChinaImages/Depositphotos
The Chinese designer and manufacturer company, Xiaomi has announced her 35th adjustment this year, in order to reduce the burden on the international business department to reorganize. Report has also revealed that Xiaomi founder, Lei Jun has withdrawn from key positions in many affiliated companies.
LatePost reported that Xiaomi staff received the 35th internal letter of the year on December 10 to change organizational structure and personnel appointments. This figure has surpassed 2020.
Lei Jun had earlier handed up a business that he directly oversaw, the business middle office department, in charge of Xiaomi's digitization and data asset building and operation.
This department has reportedly been integrated into the Group's Information Technology Department, with its head reporting to Xiaomi President Wang Xiang. This is Lei Jun's third business that he has turned over in the last month.
Youpin's e-commerce department and its software and experience department are the other two. They've been combined into the China region and the mobile phone division.
Following the restructuring, there are now seven executives who report directly to Xiaomi's CEO, Lei Jun, and are responsible for the design, production, sales, and long-term realization of Xiaomi's core business-mobile phones and IoT (Internet of Things) products.
Xiaomi has also reorganized its international business department, focusing on segmenting the overseas sales regions by region, the international network-wide e-commerce sales department, and the operator and business development department by channels, as well as the Indian business development department, which was not previously an international business department.
The regional operations department, as well as many other departments, have been combined into the international "sales department," opening up new regions and channels, and Xiang Zheng is in charge of reporting to Lu Weibing, president of the international business department.
Business Plan News Xiaomi
Yusuf is an expiring Journalist and Health law expert with a special focus on technology innovations. He is a guest writer at Qwenu and Deputy Editor-in-chief of Gamji Press.

redpajama_set_name: RedPajamaCommonCrawl


text:
Grosvenor Street Slope Compared to Other Locations Along Cycling Routes in London Ontario
by Zygmunt Gorski | Aug 20, 2022 | News
Proposed construction at the Thames Valley Parkway in the City of London has necessitated a detour route. The original detour, encompassing Talbot St and Grosvenor Street met with opposition from users who proposed an alternate route by way of Gunn Street. Initially the City of London appeared to abandon the original detour in favour of the Gunn Street route. But subsequently the City changed its mind again and indicated that both routes would be posted, allowing users to decide for themselves.
Many opposed the original route because of the danger of sending cyclists through the busy section of Talbot south of Oxford Street. This location also included a narrowing of Talbot Street as it passed through the underpass of the CP railway line just south of Oxford.
In these discussions there has been no recognition that additional dangers exist in the original detour route. In particular the original detour would cause cyclists to ride down the steep downslope of Grosvenor Street into the large parking lot at Gibbons Park. This route has been taken by cyclists in the past and there is no information about the numbers of cyclist collisions and injuries that might have occurred there. Yet, in general, there is no information that is publicly available in London as to how many cyclists become injured from collisions, where those collisions occur, and under what circumstances.
Gorski Consulting has been conducting a variety of testing in recent years on downslopes where cyclists ride in the City of London. This testing has shown that the speed of cyclists is proportional to the steepness and length of these downslopes. While speed is not the only factor in cyclist injury causation it is one of the major factors that cannot be ignored. Of particular concern at the Grosvenor Street downslope is that, upon reaching the bottom of the slope, cyclists enter into a large and busy parking lot. There is a reasonable likelihood that vehicles turning into and out of a parked position will enter into a conflict with cyclists. Cyclists are likely to be difficult to detect by drivers who may be looking in different directions when backing in or out of parking spots. The potential for increased conflicts exists when more cyclists may be directed into the parking lot as a result of the proposed detour route.
This is a view looking west along Grosvenor Street from just west of St George St. This is at the top of the downslope that would carry cyclists into the parking lot of Gibbons Park approximately 200 metres west of St. George Street. Although there is a warning sign suggesting the users of the road keep to a speed of 20 km/h, such an advisement is rarely followed, especially by cyclists whose actions are different from drivers of motor vehicles.
View looking west on Grosvenor Street from approximately halfway down the slope. The portion of the road leading up to the parking lot in the background is the steepest portion of the slope reaching a maximum of about 15.7 % at 175 metres.
View looking east from the Gibbons Park parking lot showing the end of the Grosvenor Street downslope in the background. There is a concern that, with higher numbers of cyclists entering into this area at high speed there could be a conflict with drivers of motor vehicles entering and exiting their parking spots.
Gorski Consulting has conducted some measurements of the downslope of Grosvenor Street between St George Street and the parking lot at Gibbons Park. The distance between these two endpoints is about 200 metres. The slope measurements are noted in the table below.
In comparison the slope at several sites where previous testing has been completed are noted below:
Meadowlily Rd: Distance = 400 metres, Average Slope = 4.99 %, Maximum Slope = 7.34 %
TVP at Trafalgar: Distance = 300 metres, Average Slope = 3.90 %, Maximum Slope = 9.5 %
TVP W of Richmond: Distance = 80 metres, Average Slope = 6.40 %, Maximum Slope = 11.6 %
Speeds of a Trek, hybrid bicycle, were documented as the cycle was coasted from a stopped position. The maximum speed at each of the above sites is noted below:
Meadowlily Rd Maximum Coasting Speed = 42.4 km/h
TVP at Trafalgar Rd Maximum Coasting Speed = 39.1 km/h
TVP W of Richmond St Maximum Coasting Speed = 30.3 km/h
Maximum speeds were also documented of cyclists riding on these slopes as noted below:
Meadowlily Rd Maximum observed speed = 62.72 km/h
TVP at Trafalgar Maximum observed speed = 52.94 km/h
TVP at Richmond Maximum observed speed = 43.37 km/h
If cyclist speeds, such as those noted above, occur at the Grosvenor site then there should be reason for concern. It can be noted that the steepest slope at the Grosvenor site occurs in the vicinity of 175 metres and this is just as cyclists would be approaching the parking lot. With the recognition that they are entering into a parking lot cyclists may apply higher braking at the Grosvenor site. Alternatively there could be cyclists entering into this area who have not ridden down the slope before or have limited experience with it. They might also not comprehend the danger or riding into a parking lot at high speed. Thus there is a reason to conduct testing to explore this potential safety issue.


As a FBA (Fulfillment by Amazon) seller on Amazon, we are bound by the Amazon return rules as they are the ones sending out the product; we don't handle shipping or returns for any of our products sold on Amazon. Please go through the return process on your Amazon account and you should be able to see the order and the status.
MopTop stands behind our products 100% and in conjunction with Amazon, we want our customers to have exactly what they ordered and be completely satisfied.


Studying viral transmission in hatchery-reared fish presents unique challenges. Author Rachel Breyta, now at Oregon State University, led the study while a Postdoctoral Fellow at the Cary Institute. She explains, "While hatcheries can be disinfected and controlled, juveniles migrate to the Pacific Ocean as part of their life cycle. When they return to hatcheries to spawn as adults, complicating variables - such as swimming through IHNV-infested waters - come into play."
To prevent IHNV outbreaks, many hatcheries rely on water security - using water sources at low risk of being IHNV-contaminated, such as a spring or well water, and biosecurity - disinfecting eggs with iodine to kill the virus. Despite these measures, IHNV persists, partly because many older conservation hatcheries do not have access to secure water and operate with river or creek water that contains fish that are potential IHNV hosts.


Compare cell phone carriers in Smithers, WV to find the best cell phone carrier for your needs using PhoneDog. Compare cell phone carrier coverage maps in your area. Click or tap on the wireless carrier logos to view the coverage map for that carrier. Use our cell phone plan comparison tool to find and compare the wireless carrier plans, pricing, and more. The comparison tool includes major and minor carriers. The minor carriers and MVNOs run on the networks of one of the major carriers.
When shopping for a cell phone plan in Smithers, WV, the options can be overwhelming. The diverse array of providers out there offer different plans, rates, and coverage. How do you know that a cell phone plan is right for you? How do you know what you will really need? How much money should you realistically spend? Many attractive deals will attempt to seduce you with attractive deals on talk, text, and data limits, but you can't know whether they're just after your wallet. That's why it's important to compare each coverage plan out there, scooping up as much as you can learn about the cell phone options out there and sifting through the different data, text, and talk plans until you've finally found the perfect one. At PhoneDog, we provide that information for you, letting you weight the different provider options with ease, saving you money, and helping you find the right price.
When choosing a carrier provider, it's always important to assess the coverage options in your area. Before picking a cell phone provider in Smithers, WV, you should make sure that your area has coverage. You wouldn't want to commit to a contract, only to go home and discover that you lack signal reception in your home, creating an uncomfortable situation where all your calls go straight to voicemail, making you have to leave the house every time you want to make a call. Even if you are paying relatively little for a cell phone plan, if you lack coverage in your area, your money will have been completely wasted. However, most carriers cover most areas of the country very well these days, meaning that you shouldn't have anything to worry about. Nevertheless, one should also check and double check your service carrier provider's coverage maps to make sure that your city is covered.
The citizens of Smithers, WV, and everywhere else, look for similar things in a cell phone provider. You will want the maximum amount of minutes, adequate amount of text messages, and a data plan designed for your information gathering needs. Our services display this information for you all at once, letting you compare the rates of different providers, giving you everything you need to custom tailor a plan that will save you the most money. When choosing a plan, you need to consider data limits, available minutes, and the text message rates or quotas.
Most people depend heavily in Smithers, WV on their cell phone throughout the day, so it's important to get the right coverage plan. If you frequently text, are constantly on the phone, or like to look up information without a wireless connection, you should take these factors into account when choosing a provider.
On the other hand, if you have a family in Smithers, WV, you can often save money by grouping your costs together into a single package. While the monthly payments are often higher, the shared costs leads to reduced payments. This lets you have a cell phone plan that fits each member of your family. Such plans are usually the best deal if you are looking to pay for two or more people's phone bills. This lets you share minutes, texts, and data usage, and should be planned in order to fit everyone in the family's needs. If you have kids, you should take into account that they are more likely to text and use data on their phones, meaning that if you don't set the data and text limits high enough, they could quickly run up your bill before you realize it. For this reason, it's a good idea to have a large blanket of coverage, preventing you or your family from accidentally going over, as these data charges are typically far from cheap. Make sure that you anticipate your possible usage, or you could be committed to a non-optimal data plan for the foreseeable future.
Contract plans typically make you commit to two years, although some only run for a single year. Payments are usually made on a monthly basis, and you will not be able to break you contracted without an expensive fine, usually in proportion to the remaining time left on the contract. Breaking a cell phone contract can often be expensive, which is why you should always try to make sure that you get the best plan available in Smithers, WV.
To choose the right plan in Smithers, WV, you should take the time to determine what your needs are. How many people will be on your plan? How many devices do you have? How much data do you plan to use? How many texts and calls will you be sending or receiving? Talk, text and data are often the most important factors to look at when deciding on a plan. These are the things most advertised by cell phone carriers.
Data usage has recently become a major factor when choosing a cell phone plan. With the development of smartphones, almost everybody with a cell phone uses data. Most cell phone carriers in Smithers, WV charge a large amount of money for data, and it is hard to find a truly unlimited plan. In most cases, a plan will say that unlimited data is offered even though there are hidden technicalities that keep you from actually having unlimited data. Usually, this is the case regarding high-speed data. Unlimited plans are starting to become more popular, however, so they may be easier to find in the near future.
What's The Best Cell Phone Carrier in Smithers, WV?
All four of the following carriers offer coverage in Smithers, WV: AT&T, Sprint, Verizon Wireless, and T-Mobile. There are even smaller companies, called MVNOs, which conduct their service under one or more of these four carriers. Your city is more than likely covered by one or all of these carriers.
Sprint is known for being one of the most affordable national cell phone carriers. They provide many types of plans with service in Smithers, WV, including an affordable unlimited one.
Verizon Wireless, another popular choice in Smithers, WV, offers family sharing, no contract, and individual plans. You can pick from an unlimited plan, shared data plan or a prepaid plan when you sign up. A shared data plan can be used by one individual or within a family and comes with unlimited talk and text. It also comes with a predetermined amount of data.
MVNOs, the aforementioned cell phone carriers, are another choice that can be compared with our comparison tool. These carriers are basically a sub-division of the national carriers and usually offer many deals for their plans which typically offer service in Smithers, WV as well.


text:
A NEW MEDIA ATTORNEY DISCUSSES FACEBOOK'S TOS
by Scott Macaulay
I posted below about the change in Facebook's Terms of Service and the uproar it has caused online. From reading the various articles and postings, it seems like there are two groups of people discussing this. First, there are the regular Facebook users who employ the service to keep up with the friends, swap links, and occasionally post personal photos that are never intended to have a commercial use. Some of these people are disturbed by Facebook's policies because they don't want their daughter to grow up and become the next Angelina Jolie and then find all of her high-school pictures plastered on top of Facebook ads or, worse, sold to the tabloids. Others simply don't care or don't anticipate enough value accruing to their personal photos, private messages and Wall postings to worry about it.
Then there is the group consisting of creative professionals and serious artists who see Facebook as a way to spread the word about their work and keep their fans updated. In order to do this, they may post photos, texts, films, or excerpts of larger works as a way to promote themselves or perhaps just share for no remuneration what they've been working on. Some of these people may hope for a big payday someday. Others might subscribe to a gift economy ethos and not want to see money change hands for their work. And others may be in between, sharing some work while still hoping to get paid down the line. For all of these people, the broad license that is seemingly granted to Facebook is a big issue. Yes, as Mark Zuckerberg has written, Facebook is in the business of creating an inviting space, and exploiting its members' original content would not be wise business. But, any artist knows that when it comes to intellectual property, one must assume that what is allowed by a written contract could one day come true.
For an analysis of what's in that written contract, here is a post and a short podcast for an attorney at new media law firm Jacobson Attorneys. It's from their web.tech.law blog.
Remainders By Nicholas Rombes on Nov 15, 2013
Cinekink Announces its 2013 Line-Up By Scott Macaulay on Feb 12, 2013
Wes Anderson on Using Throwback Ratios, Romantic Worldviews, and European Reconnaissance to Craft The Grand Budapest Hotel By R. Kurt Osenlund on Mar 8, 2014
Gotham Award Nominations Announced: Beasts, Bernie, Middle and Moonrise Lead the Way By Nick Dawson on Oct 18, 2012


The International Federation of Fertility Societies (IFFS) and its 2019 World Congress Scientific Committee are honored to announce the 23rd IFFS World Congress, to be held April 11-14, 2019 in Shanghai, China. Themed "Transforming the Frontiers of Human Reproduction" the Congress will highlight contributions from renowned international faculty focused on delivering leading edge research across a range of current issues in our field.
International co-operation and dialogue in fertility and treatment can be said to have truly commenced in 1951 in Rio de Janeiro with the formation of the International Fertility Association (IFA). Eminent scientists from around the world became members and the triennial World Congress in Fertility and Sterility was organized and held in New York City, USA, in May, 1953. At the Sixth World Congress (1968) IFFS was formally organized.
For the past fifty years, the IFFS World Congress has been a hallmark of global reproductive healthcare. We have distinguished ourselves in service to our core mission: to emphasize cutting edge basic and translational science; to provide education to practitioners globally; and to promote high standards of care and advocating for patients.
The 2019 World Congress will examine research and development on local, regional, and international levels. We sincerely hope you will be able to join us in Shanghai, where 4,000 fellow international specialists will continue to carry this legacy forward.


text:
Brian Kemp has been caught Red-handed Suppressing the Right to Vote
Retiree Organization Calls on him to Resign Immediately as Secretary of State
The following statement was issued by Kenny Bradford, President of the Georgia Alliance for Retired Americans, following Brian Kemp's comments about exercising one's right to vote, first publicized in Rolling Stone:
"The Secretary of State's private comments, which show that Brian Kemp is 'concerned' about Georgians exercising their constitutionally guaranteed right to vote, should be deeply disturbing to anyone who cares about a level playing field.
"Voter suppression takes many forms, but it always disproportionately affects seniors, minorities and the disabled.
"The 65,000 members of the Georgia Alliance for Retired Americans call on Secretary Kemp to resign as the Georgia Secretary of State immediately. We simply do not have confidence that he will impartially and fairly administer this election.
"The right to vote is sacred and has been hard earned. Brian Kemp's words and deeds have shown that he cannot be trusted to fulfill his job as our state's chief election administrator."
Contact: Kenny Bradford, 770-971-3550 or kbrad.021651@gmail.com
Written by rebecca
Topics: Press Releases


text:
Edwin the Osprey's Last Migration
Published by Connecticut State Environmental Conservation Police
April 24, 2016 at 7:06pm ·
Recently one of our Officers assigned to the Eastern Sector of the Marine District was on patrol at Miller Pond in Waterford when he came across an osprey with a very interesting story to tell.
The officer learned that the osprey he located was a seven-year-old male named "Edwin" who was outfitted with a leg band and a solar-powered satellite transmitter. After doing some research the Officer learned that Edwin was tagged on May 11th 2013 on Fishers Island, NY by a Wildlife Biologist and Biology Professor at The University of North Carolina who was conducting research on the migration patterns of Ospreys. Edwin is an osprey who calls Connecticut home, specifically the East Lyme/ Waterford area.
On August 16th Edwin departed Connecticut and began his annual migration south. Biologists were initially concerned as Edwin was migrating nearly a month earlier than normal. Nevertheless, Edwin travelled south along the east coast through Florida, stopping occasionally in Cuba, Florida Keys and the Dominican Republic. Edwin continued on south through the Caribbean Sea and toward South America travelling through both Columbia and Venezuela. Edwin's transmitter lost service for several weeks as he travelled through Venezuela, but he was relocated on Valentine's Day as he travelled past the Rio Solimoes (upper branch of Amazon River). Edwin spent his winter months along the Amazon River and Solimoes River located in the northern and western regions of Brazil. Edwin returned to Connecticut on April 2nd after having traveled over 5,000 miles.
Unfortunately, this migration was Edwin's last as he passed away shortly after his return to Waterford. An average osprey lifespan is anywhere from five to ten years of age, and in terms of Osprey years, Edwin was an old bird. An old bird with an amazing story.
For more information on Edwin's life and migration you can visit http://fergusonmuseum.org/osprey-cam/track-edwin/ and http://ospreytrax.com/OspreyMainPage.html.
From ctpost.com/local | By Jim Shay | Thursday, April 28, 2016
Conn Osprey Dies after Long Journey from Amazon


text:
Knox County Student Honored As 2009 Girl of Distinction By Girl Scouts of Eastern MO
Tara Mason, a senior at Knox County High School, has been selected as a 2009 Girl of Distinction in the category of Government & Legal.
Girl Scouts of Eastern Missouri (GSEM), the premier leadership organization for girls, is honoring ten girls and ten women in ten different categories: Creative Arts, Entrepreneurship, Government & Legal, Inspiration, Numbers, Practical Arts, Social Sciences, Sports & Fitness, Technology and The Sciences.
Mason participated in the United States House of Representatives Page Program in Washington, D.C. She has been involved with the Youth Voter Participation Program, served as an election judge, and will be job shadowing the assistant prosecuting attorney for her district.
"We are proud to honor an outstanding group of female leaders who are making a positive difference in our community," said Marcia Barber, CEO of Girl Scouts of Eastern Missouri. "The 2009 Girls and Women of Distinction represent a diversity of impressive accomplishments and reflect the principles of Girl Scouting."
The 20 award recipients will be honored at the Girls and Women of Distinction awards reception on Tuesday, April 21, at Old Warson Country Club in St. Louis, Mo.
AmerenUE is the event's Title Sponsor. June McAllister Fowler serves as Event Chair. Donna Martin is Honorary Chair.
For more information about the Girls and Women of Distinction event, please contact Marcy Bursac at 314.592.2367 or mbursac@girlscoutsem.org.