"Ergodicity" of non-invariant product measures (with respect to the shift)

Question

Let $(X_n)$ be a sequence of independent but not necessarily identically distributed random variables with $\mathbb{E}X_n = 0$ for all $n$. If the $X_n$ are uniformly bounded, Kolmogorov's strong Law of Large Numbers implies that $(1/n)\sum_{i=1}^n X_i \rightarrow 0$ almost surely.

Now let $S$ be a metric space and let $T$ be the shift operator on $S^\mathbb{N}$, that is $T(x_0,x_1,\dots) = (x_1, x_2,\dots)$. Consider a product probability measure $\mu = \prod_{i=0}^\infty \mu_i$ on $S^\mathbb{N}$ (thus $\mu$ is generally not invariant unless $\mu_i = \mu_0$ for all $i$).

For a given bounded, continuous function $f$ on $S^\mathbb{N}$ of the form $f = g \circ \pi$, where $g$ is a continuous bounded function on $S$ and $\pi(x_0,x_1,\dots) = x_0$, let $$X^f_n(\boldsymbol{x}) := f\circ T^n(\boldsymbol{x}) - \int f\circ T^n(\boldsymbol{y})\,\mu(\mathrm{d}\boldsymbol{y}).$$ On the probability space $(\Omega, \mathbb{P}) := (S^\mathbb{N}, \mu)$, we have that $(X_n^f)$ is a sequence of independent random variables with $\mathbb{E}X_n^f = 0$ for all $n$, with $\sup_{\boldsymbol{x},n}X_n^f(\boldsymbol{x}) < \infty$. Thus by the SLLN we have $(1/n)\sum_{i=0}^{n-1}X_n^f \rightarrow 0$ ($\mu$ a.e.).

Therefore, given a product probability measure on $S^\mathbb{N}$, a property analogous to ergodicity (w.r.t. $T$) holds when we restrict ourselves to continuous bounded maps of the form $f = g\circ\pi$ as above.

My question is: does this property still hold for the general $f\in C_b(S^\mathbb{N})$?

Answer 1

No. For example, take $S=\mathbb R$ and let $\mu_n$ be Lebesgue measure restricted to $[n,n+1]$. Set $$ f(x_0,x_1,\ldots)=\sin^2(2\pi x_0)\sin(2\pi(x_{2^j-\lfloor x_0\rfloor}) \text{ if $x_0\in [2^{j-1},2^j)$}. $$

Then $f(T^n(x_0,x_1,\ldots))=\sin^2(2\pi x_n)\sin(2\pi x_{2^j})$ for all $n$ in the range $[2^{j-1},2^j)$ (the reason for the $\sin^2$ term is just to make it continuous). The averages don't converge.