Background

For a topological space $X$, let $T: X \to X$ be a continuous application. Then, call the set of $T$-invariant probability measures $M(T)$, and call the set of $T$-ergodic (probability) measures $E(T)$. It is evident that $E(T) \subset M(T)$. But it may happen that $M(T) = \emptyset$. For example, take $X = \mathbb{R}$ and $T(x) = x+1$.

Since an ergodic measure is invariant, it is immediate that

$ \begin{equation*} \sup_{\mu \in E(T)} h_\mu(T) \leq \sup_{\mu \in M(T)} h_\mu(T). \end{equation*} $

The question is whether equality holds or not. When $X$ is compact, it is well known that equality holds. In this case, it is a consequence of Jacobs' Theorem, which states that for any $\mu \in M(T)$, there exists a measure $\tau$, over the set $E(T)$, such that

$ \begin{equation*} h_\mu(T) = \int_{E(T)} h_m(T) d\tau(m). \end{equation*} $

When $X$ is compact (locally compact, in fact), the above equation is a consequence of Choquet Representation Theorem and the Krein-Milman Theorem. (See, for example, Theorem 8.4 from Walters, P. An Introduction to Ergodic Theory)

Now, when $X$ is not necessarily compact, but it is a Borel subset of a compact metrizable set $\widetilde{X}$, Pesin and Pitskel' argue in their Topological Pressure and the Variational Principle for Noncompact Sets, at the end of page 310: (I will rename the spaces and applications in order to conform to this post's notation.)

We may assume that measure $\mu$ is ergodic. In fact, consider the partition $\eta$ of $X$ into ergodic components $X_s,\, s \in S$, of measure $\mu$. Denote by $\mu_s$ the measures on $X_s$ (then $T * \mu_s = \mu_s$), and by $\nu$ the measure on the quotient space $X / \eta$. Then $h_\mu(T) = \int_{Y/\eta} h_m(T) d\nu(m)$.

As far as I understand, $Y/\eta$ is just the same as $E(T)$, since each ergodic component is associated with an ergodic measure. And for the same reason, $\nu$ is just our $\tau$. So, what is being stated is the validity of

$ \begin{equation*} h_\mu(T) = \int_{E(T)} h_m(T) d\tau(m), \end{equation*} $

which in turns implies the equality

$ \begin{equation*} \sup_{\mu \in M(T)} h_\mu(T) = \sup_{\mu \in E(T)} h_\mu(T). \end{equation*} $

In Pitskel' and Pesin's paper, $T$ is not even supposed to be the restriction to $X$ of a continuous transformation $\widetilde{T}: \widetilde{X} \to \widetilde{X}$.

Questions

1. How do I prove that when $X$ is a Borel subset of a compact metrizable space $\widetilde{X}$ and $T$ is a continuous application $T: X \to X$, then for any $\mu \in M(T)$, there exists a measure $\tau$ over $E(T)$ such that $h_\mu(T) = \int_{E(T)} h_m(T) d\tau(m)$?

2. In case the answer to question "1" is negative, is there a prove for the specific case where $T$ is the restriction of a continuous application $\widetilde{T}: \widetilde{X} \to \widetilde{X}$?

3. Do you know nice examples of transformations of measurable spaces where $E(T) = \emptyset$ while $M(T) \neq \emptyset$?

PS: This is my first post to MathOverflow. This is really exciting! :-)