Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the set of all ergodic measures, exactly the extremal points. Moreover, the set $\mathrm{C}(\mathcal{M}^e)$ of convex combinations of elements of $\mathcal{M}^e$ is dense in $\mathcal{M}$ (in the weak topology).


What is known about which measures in $\mathcal{M}$ are (and are not) in $\mathrm{C}(\mathcal{M}^e)$? In particular, what is an example of a measure in $\mathcal{M} \setminus \mathrm{C}(\mathcal{M}^e)$? What about just a measure in $\mathcal{M} \setminus \mathcal{M}^e$?


I've tagged with 'reference-request' because, besides just answers to the above questions, I'd like some references to where this stuff has been studied.



Shift invariant measures that are(n't) convex combinations of ergodic measures
The ergodic decomposition theorem states that any shift-invariant Borel probability measure, $\mu$, can be *uniquely* expressed in the form
$$
\mu=\int\_{\mathcal M^e(X)}\nu\,d\eta(\nu).
$$


This means that $C(\mathcal M^e)$ is the set of measures that are (finite) convex combinations of ergodic measures and $\mathcal M\setminus C(\mathcal M^e)$ consists exactly of those measures, for which the measure $\nu$ appearing in the decomposition is not finitely supported.


Any non-trivial combination of ergodic measures is non-ergodic.


For a measure that is not a convex combination of ergodic measures, you need to take a measure that is a combination of an infinite number of measures.


For simple examples, let $\mu\_\beta$ be the Bernoulli measure on $\lbrace 0,1\rbrace^{\mathbb Z}$ where in each coordinate the symbol 1 appears with probability $\beta$ independently of all other coordinates.


Then $\frac12(\mu\_{1/2}+\mu\_{1/17})$ is an element of $C(\mathcal M^e)\setminus \mathcal M^e$. A "typical point" from this measure is either (with probability .5 a string of 0's and 1's with a 50% density of 1's'; or with probability .5 a string of 0's and 1's with a 1/17 density of 1's).


The measure $\mu=\int\_0^1 \mu\_\beta\,d\beta$ (the uniform mixture of all Bernoulli measures) belongs to $\mathcal M\setminus C(\mathcal M^e)$. If you pick a typical point from this measure, there is some density $\beta$ (which itself is uniformly chosen from 0 to 1) and then the sequence of 0's and 1's has density $\beta$ of 1's.


