My question is about the structure of the set of infinite Borel measures on compact metric spaces invariant with respect to a homeomorphism.

Let $T$ be a homeomorphism of a compact metric space $X$ and $M(T)$ the set of $T$-invariant Borel measures on $X$, i.e., for any $\mu$ from $M(T)$ one has $\mu(TA) = \mu(A)$ for any Borel set $A$. Denote by $M_1(T)$ the subset of probability Borel measures and let $M_\infty(T)$ denote the subset of infinite $\sigma$-finite measures from $M(T)$. Remark that the set $M_1(T)$ is always non-empty; on the other hand, the set $M_\infty(T)$ may be empty in some cases. For instance, if $T$ is minimal (every $T$-orbit is dense in $X$), then $M_\infty(T)= \emptyset$. It is well known that $M_1(T)$ is a Choquet simplex whose extreme points are ergodic measures.

**Question**: What can be said about $M_\infty(T)$? Is this set a Choquet simplex?

(I'm sorry if this is trivial or well known)