Good afternoon.
Can anybody give me an example of a continuous map $T:X\to X$ defined on a Polish space $X$ which admits an invariant Borel probability measure but no ergodic Borel probability measure? Typically, the space $X$ could be a separable infinite-dimensional Banach space, or the space $\mathbb N^{\mathbb N}$.
Thanks in advance.
The existence of such an example is prevented by the ergodic decomposition theorem, which asserts that every $T$-invariant measure on a standard probability space $(X,\mathcal{B},m)$ can be expressed as a (possibly uncountably infinite) convex combination of ergodic $T$-invariant measures by means of an integral over the set of ergodic $T$-invariant measures on $(X,\mathcal{B},m)$. In particular, since the integral must have a nonzero outcome the set of such measures is nonempty.
This theorem is relatively technical to prove and seems to be left unproved or even unstated in standard textbooks on ergodic theory. For example, Walters (p.34) describes the theorem without stating it formally, referring instead to the original work of V. I. Rokhlin (Selected topics in the metric theory of dynamical systems, Uspekhi Mat. Nauk. 4 (1949) 58--127). Petersen (p.81) mentions the result in passing but does not provide a reference. Aaronson's book on infinite ergodic theory gives a proof of the ergodic decomposition theorem for probability spaces in the case where $T$ is invertible (p. 62--64) and sets the case of general $T$ as an exercise; Aaronson attributes the result to von Neumann (who discovered it independently of Rokhlin) and cites Zur Operatorenmethode in der Klassischen Mechanik, Ann. Math. 33 (1932) 587--642.
Terence Tao's weblog has a nice discussion of the result here.
