# On the set of infinite measures

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)

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I think if you take a sum of $\delta$ measures along any (aperiodic) orbit then you always get an infinite measure. Not a nice one because it's not locally finite. – Anthony Quas Jun 22 '12 at 20:11
Actually I think just about everything goes wrong. There's a nice topology on the set of probability measures, thanks to the Riesz representation theorem. The same isn't true for the infinite measures. The probability measures have a natural normalization (so that only convex combinations make sense). On the other hand, a positive multiple of an infinite invariant measure is another infinite invariant measure. – Anthony Quas Jun 23 '12 at 8:11
Anthony, thanks for your comments. You are right, infinite atomic invariant measures always exist. Of course, the case of continuous infinite measures is of main interest. Do you know whether there are any results about erodic decomposition of dynamical systems with infinite invariant measures? – SIB Jun 23 '12 at 14:12