What is known about the space of measure-preserving transformations? I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations.
The books that I have on ergodic theory (P. Walters, K. Petersen) don't really address that, and I've only seen one reference to the Halmos book, where he supposedly defines a metric on this space. But is this, for example, a Banach space? I would assume it's not, but I'd like to hear more on it and where I could get some information on this.
 A: Let me mention some related results.
Given a compact metric space $X$, the set $Y$ of all Borel probability measures on $X$ is metrizable and in fact the induced topology on $Y$ makes it compact.
The first property is a more or less easy consequence of the separability of the space $C(X)$ of continuous functions on $X$ with the supremum norm, which allows us to define a distance on $Y$ by
$$
d(\mu,\nu)=\sum_{n=1}^\infty\frac1{2^n}\left|\int_X\phi_n\,d\mu-\int_X\phi_n\,d\nu\right|,
$$
where $\phi_n$ is any fixed sequence of continuous functions whose closure (that is, the closure of their union) is the closed unit ball in $C(X)$.
Note that the induced notion of converge is simply weak convergence.
The compactness of $Y$ has in particular the following sequence: any continuous map on a compact metric space has at least one $T$-invariant probability measure. So, under that hypotheses, the subset of all $T$-invariant measures in $Y$ is a nonempty compact convex set (that has either one element of infinitely many elements). Incidentally, the extreme points of the convex set are precisely the ergodic measures.
The general problem of describing the set of invariant measures in other contexts really depends on he hypotheses and to the best of my knowledge there exists no general theory.
