# 21. Mixing from a Measure-Theoretic Viewpoint

In this lecture we define what it means for a measure-preserving dynamical system to be *mixing *and *weakly mixing*.

The definition of mixing is very natural: if $f$ is a measure-preserving dynamical system on $(X,\mathcal{A}, \mu)$ then $f$ is mixing if for all measurable sets $A,B$ one has

$$

\lim_{k \to \infty} \mu(f^{-k}A \cap B) = \mu(A)\mu(B).

$$

The definition of weak mixing is rather less intuitive. For this we require that for all measurable sets $A,B$ one has

$$

\lim_{k \to \infty} \frac{1}{k} \sum_{i=0}^{k-1}\big| \mu(f^{-i}A \cap B) - \mu(A) \mu(B)\big| = 0.

$$

Perhaps even more confusingly, at first glance this condition appears to have nothing to do with the definition of weak mixing in the topological world (see Lecture 5).

This confusion will be rectified by the end of the lecture, when we prove—consistently with the topological definition—that $f$ is weakly mixing if and only if $f \times f$ is ergodic.

We conclude with the measure-theoretic analogue of Furstenberg's Theorem from Lecture 6.

Comments and questions?