First I will present two definitions. Let $f:X \to X$ a continuous measurable transformation.
- A probability measure $\mu$ defined on the Borel sets is said to be $f$-invariant if $\mu(f^{-1}(A)) = \mu(A)$ for every Borel set $A$.
- A map $f$ is said to be mixing with respect to some $f$-invariant probability measure $\mu$ if $$|\mu(f^{-n}(A) \cap B) - \mu(A)\mu(B)| \to 0, \qquad\text{when } n \to \infty,$$ for any measurable sets $A,B$.
A natural question that arises is the speed of mixing. In some books and articles I see some variation of the phrase: "It is possible to find subsets $A, B$ such that the convergence in the definition of mixing is arbitrarily slow."
When talking about speed of mixing, I'm talking about a sequence $a(n) \downarrow 0$, such that
$$|\mu(f^{-n}(A) \cap B) - \mu(A)\mu(B)| \le Ca(n), \qquad\text{for all } n.$$
Thus, what the phrase means that if $f$ is mixing then given a sequence it is possible to find sets $A,B$, so that the inequality above is not true for all $n$.
I haven't found a proof for this claim, just specific examples. What I managed to do was prove this statement when I use the definition of mixing with correlation function (see definition below) in $L^2$. For this, I used the Riesz representation theorem.
For the case above I couldn't prove it. I also don't know if it is necessary to make any assumptions about the space $X$, in this case we only have $\mu(X)=1$. I would appreciate an idea for the proof or a reference.
Definition of mixing with functions:
For measurable functions $\varphi,\psi : X \to \mathbb{R}$ we define the correlation function $$C_n(\varphi,\psi) =\left|\int \psi(\varphi\circ f^n)~d\mu - \int \psi ~d\mu \int \varphi~d\mu\right|.$$We say that $f$ is mixing if $C_n(\varphi,\psi) \to 0$ as $n\to\infty$.