Measure Preserving Transformations and Ergodicity

Does the proof of the Bogoliubov-Krilov Theorem also provide a proof in the case $X$ is compact Hausdorff but not necessarily metrizable?

Let $(X, \mathcal{M}, \mu )$ be a finite measure space and $T: X \rightarrow X$ a measurable transformation. For a measurable function $g$ on $X$, define the measurable function $U_{T}(g)$ by $U_{T}(g)(x) = g(T(x))$. Show that $T$ is measure preserving if and only if for every $1 \leq p < \infty , U_{T}$ maps $L^{p}(X, \mu )$ into itself and is an isometry.

Suppose that $T: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is linear. Establish necessary and sufficient conditions for $T$ to be measure preserving with respect to Lebesgue measure on $\mathbb {R}^{n}$.

Let $S^{1} = \left\{ z = e^{i \theta } \mid \theta \in \mathbb {R}\right\}$ be the circle with the group operation of complex multiplication and $\mu$ be Haar measure on this group (see Problem 24). Define $T: S^{1} \rightarrow S^{1}$ by $T(z) = z^{2}$. Show that $T$ preserves $\mu$.

Let $T$ be a measure preserving transformation on a finite measure space ($X, \mathcal{M}, \mu$) and the set $A$ be measurable. Show that for almost all $x \in X$, there are infinitely many natural numbers $n$ for which $T^{n}(x)$ belongs to $A$.