## 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_)

[Choquet Representation Theorem]: "http://en.wikipedia.org/wiki/Choquet_theory"
  (Theorem about generalized convex combinations)
[Krein-Milman Theorem]: "http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem"
  (Theorem on existence of extremal points)

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! :-)