Stochastic modelling, distribution and ergodicity of a particular time series with a given finite history Let $\Omega$ be a sample space. A stochastic process $\{Y_t\}$ is a function of both time $t \in \{1, 2, 3, \ldots\}$ and outcome $\omega \in \Omega$.

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*For any time $t$, $Y_t$ is a random variable (i.e. a function from $\Omega$ to the space of real numbers $\mathcal{R}$).

*For any outcome $\omega$ the series $y(\omega)$ is a time-series of real numbers: $\{y_1(\omega), y_2(\omega), y_3(\omega), \ldots \}$
We can model a particular time series with an observed history $(y_1,\dots,y_t)$ as a realization of a stochastic process.
It is quite common in practice* that

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*we are only interested in a particular observed series (corresponding to a particular $\omega_0\in\Omega$) and its future or historical development and

*we do not care about the hypothetical other realizations of the same stochastic process (corresponding to $\omega\in\Omega\backslash\omega_0$).

I.e. we want to make inference and predictions about $\{y_t({\omega_0)}\}$ rather than $\{Y_t\}$ and/or $y_{t+1}({\omega_0)}$ rather than $Y_{t+1}$. But then it seems to me we cannot talk about the distribution of $\{y_t(\omega_0)\}$ (a realization of a random process) or $y_{t+1}(\omega_0)$ (a realization of a random variable) or any derived features such as moments. Yet time series analyses are full of expected values (conditional or unconditional), variances (cond. or uncond.) and the like. And then there are discussions of ergodicity which again seem irrelevant if we only have a single, fixed $\omega_0$. I am confused by this. How do I think about it?
*An example: the daily closing share price of Tesla. We can obtain the historical time series $(y_1,\dots,y_t)$ from a financial database. We may want to discover patterns in the series and extrapolate them into the future, perhaps with an aim of investing in Tesla's share and making money or (more modestly) assessing and managing risk. We do not care about any other realization from the data generating process (DGP) has generated Tesla's daily share price, as we do not believe there exists another company the share price of which is governed by the exact same laws.
A counterexample: second-by-second altitude of a helium atom in a closed container filled with pure helium. Roughly, every atom is the same, and the laws that govern atom A govern also atom B and all the other atoms in the container. If we built a model based on the historical time series of atom A, we could still be interested in generalizing across atoms and saying that the model applies to each of them. Ergodicity may then naturally be relevant.

A follow-up question: "Stationarity and ergodicity of a process conditional on a finite trajectory".

There can be more than one $\omega\in\Omega$ that may have generated the finite-length observed trajectory $(y_1,\dots,y_t)$. Let us collect all such $\omega$s into an $\Omega_0\subset\Omega$. By referring to the distribution of $\{Y_t\}$ or $Y_{t+1}$, are we implicitly considering its conditional distribution, i.e. the distribution over the subset $\Omega_0$. The distribution is proper if $\Omega_0$ contains more than a single element.
I am not sure how to think about ergodicity. Perhaps it is still relevant within $\Omega_0$ (as long as it contains more than a single element) even if we are not interested in $\Omega\backslash\Omega_0$.
