In the last few lectures we started with a probability space $(X, \mathscr{A}, \mu)$, and then looked at transformations $f \colon X \to X$ which preserve $\mu$. Over the next two lectures we flip this on its head. Rather than starting with the measure $\mu$ and then restricting attention to transformations $f$ that preserve $\mu$, now we will start with $f$ and look for measures for which $f$ is measure-preserving.

Such a paradigm shift is possible purely within the measure-theoretic world, but it is maximally profitable if we begin in a topological setting. Indeed, suppose we are given a topological dynamical system $f$ on a compact metric space $(X,d)$. Let $\mathscr{B}$ denote the Borel sigma-algebra. Can we find a probability measure $\mu$ on $(X, \mathscr{B})$ for which $f$ becomes measure-preserving?

If we are successful in our quest to find such a $\mu$, the dynamical system $f$ will then simultaneously be a topological dynamical system and a measure-preserving dynamical system. The benefits of this approach should be clear: we can then bring all the results from both topological and measure-theoretic dynamics to bear when studying the dynamics of $f$.

In order to have effective methods to “find” measures for which our given topological dynamical system $f$ is measure-preserving, we need to understand what properties the space of all probability measures on the Borel sigma-algebra of $X$ has. For example, does it carry a topology? If so, is it compact? In fact, the answer to both of these questions is yes: the space $\mathcal{M}(X)$ of all probability measures on the Borel sigma-algebra of $X$ is itself a compact metric space. We prove this today.