Does the "random Krylov-Bogolyubov theorem" hold in a non-skew-product setting?

Question

Informal description.

Suppose I have a dynamical system $f$ defined on the product of a compact space $X$ representing the state space of an "experimentally visible" variable and a compact space $Y$ representing the state space of an "experimentally invisible" variable. Although $Y$ is "invisible", I assume that it is equipped with an equivalence class of probability measures that - like the Riemannian volume measure on a compact smooth manifold - defines "physically accessible" sets of initial conditions in $Y$. I do not assume that $f$ has any skew-product structure, meaning that there is allowed to be bi-directional feedback between the visible variable and the invisible variable. A heuristic interpretation of my question below is: If in my experiments I consistently observe the same ergodic statistics for the visible variable, is it necessarily a "physical possibility" that this ergodic statistics comes from an underlying stationary dynamics for the process as a whole?

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Precise formulation.

Let $X$ and $Y$ be compact metric spaces, let $\pi_X \colon X \times Y \to X$ be the first coordinate projection, let $\mu$ be a Borel probability measure on $X$ of full support, let $\lambda$ be a Borel probability measure on $Y$ of full support, and let $$ f \colon X \times Y \to X \times Y $$ be a homeomorphism. Suppose that for every continuous function $g \colon X \to \mathbb{R}$, $$ \mu \otimes \lambda \left( (x,y) \in X \times Y \, : \, \frac{1}{N} \sum_{n=0}^{N-1} g(\pi_X(f^n(x,y))) \to \int_X g \, d\mu \ \text{ as } \, N \to \infty \right) = 1. $$ Does it follow that there exists an $f$-invariant probability measure on $X \times Y$ whose $X$-marginal is $\mu$?

If so, does there necessarily exist a Borel probability measure $\nu$ on $X \times Y$ with $\lambda$-absolutely continuous $Y$-projection such that $\frac{1}{N} \sum_{n=0}^{N-1} f^n_\ast\nu$ has a subsequence converging weakly to a measure whose $X$-marginal is $\mu$?

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Remark. In the case that $f$ has the skew-product structure $f(x,y)=(\theta(x),\varphi_x(y))$ [in which case the measure $\lambda$ is irrelevant for the first question], an affirmative answer to the first question is given by Corollary 6.13 of Hans Crauel, Random Probability Measures on Polish Spaces - although quoting that result is probably overkill, and I think an affirmative answer to both questions is provided (with $\nu=\mu \otimes \lambda$) by just slightly adapting the regular proof of the Krylov-Bogolyubov theorem applied to $f$. In terms of application, typically this skew-product setup concerns the scenario that $X$ is the state space of an unknown "noise" and $Y$ is the "visible" state space - which is opposite to what I'm thinking about in my question.

Answer 1

Okay, I've seen that the proof of an affirmative answer to both questions in the general case (with $\nu=\mu \otimes \lambda$) is not very hard:

Since $X \times Y$ is compact, we can let $(k_n)$ be a strictly increasing sequence of positive integers such that $\frac{1}{k_n} \sum_{i=0}^{k_n-1} f^i_\ast(\mu \otimes \lambda)$ is weakly convergent as $n \to \infty$. Let $\mathbb{P}$ denote the limit. As in the usual proof of the Krylov-Bogolyubov theorem, $\mathbb{P}$ is $f$-invariant. It remains to show that $\pi_{X\ast}\mathbb{P}=\mu$. For any continuous function $g \colon X \to \mathbb{R}$, we have \begin{align*} \int_X g \, d(\pi_{X\ast}\mathbb{P}) &= \int_{X \times Y} g \circ \pi_X \, d\mathbb{P} \\ &= \lim_{n \to \infty} \frac{1}{k_n} \sum_{i=0}^{k_n-1} \int_{X \times Y} g \circ \pi_X \, d(f^i_\ast(\mu \otimes \lambda)) \\ &= \lim_{n \to \infty} \frac{1}{k_n} \sum_{i=0}^{k_n-1} \int_{X \times Y} g(\pi_X(f^i(x,y))) \, (\mu \otimes \lambda)(d(x,y)) \\ &= \lim_{n \to \infty} \int_{X \times Y} \frac{1}{k_n} \sum_{i=0}^{k_n-1} g(\pi_X(f^i(x,y))) \, (\mu \otimes \lambda)(d(x,y)) \\ &= \int_{X \times Y} \int_X g \, d\mu \, (\mu \otimes \lambda)(d(x,y)) \\ &\hspace{20mm} \text{by Dom. Conv. Thm. and assumption on $\mu$ and $\lambda$} \\ &= \int_X g \, d\mu. \end{align*}

QED.

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An interesting fact: I realise that there does not necessarily exist an $f$-ergodic invariant probability measure on $X \times Y$ whose $X$-marginal is $\mu$.

For example, take: $X=\mathbb{R}/\mathbb{Z}$; $Y=\{n^{-\frac{1}{2}} : n \in \mathbb{Z}_{\geq 1}\} \cup \{0\}$; $\mu=\mathrm{Lebesgue}$; $\lambda$ with $\lambda(\{y\})=0 \Leftrightarrow y=0$; and $$ f([x],y) = ([x+y] , \tilde{f}(y)) $$ for $x \in \mathbb{R}$ and $y \in Y$, where \begin{align*} \tilde{f}(n^{-\frac{1}{2}}) &= (n+1)^{-\frac{1}{2}} \quad \forall n \in \mathbb{Z}_{\geq 1} \\ \tilde{f}(0) &= 0. \end{align*}