I’m currently working on Ergodic Theory by Einsiedler and Ward, and I’m stuck at Exercise 2.7.1:
Let $(X,\mathcal{B},\mu)$ be a measure space with $\mu(X)<\infty$, and $T:X\to X$ be a measure-preserving transformation, that is, $\mu(A)=\mu(T^{-1}A)$ for all $A\in \mathcal{B}$. If for any $A,B\in \mathcal{B}$ there exists $N\in \mathbb{N}$ such that
\begin{equation}
\mu(A\cap T^{-n} B)=\mu(A)\mu(B)
\end{equation}
holds for all $n\geq N$, then it is trivial in the sense that $\mu(A)=0$ or $\mu(A)=1$ for any $A\in \mathcal{B}$.
At first I took $B\in \mathcal{B}$ such that $0<\mu(B)<1$ and tried to construct $A$ that violates the equation by using $T^{-n}B$s, but it didn’t work. Looking at the hint page, it says that such a set $A$ is obtained by applying Baire Category Theorem, but I have no ideas where to apply it. Would you please give me an extra hint?
Best Answer
This is a beautiful exercise. So I will spell out the solution suggested by the hint.
As noted by John Griesmer, the metric space $(M,d)$ of equivalence classes (up to measure 0) of measurable sets, with the metric $d(A,B)=\mu(A \Delta B)$, is complete. Indeed, $(M,d)$ is isomorphic to a closed subspace of $L^1(\mu)$.
Suppose (aiming for a contradiction) that there exists a set $B \in \mathcal B$ with $0<\mu(B)<1$. For each $n \ge 1$, the set $$G_n:=\{A \in \mathcal B: \mu (A \cap T^{-n} B) \ne \mu(A)\mu(B) \}$$ is clearly open in $(M,d)$. Thus for $k \ge 1$, the set $H_k=\cup_{n=k}^\infty G_n$ is also open.
Next, we show that $H_k$ is dense. Given a set $A \notin H_k$ with $\mu(A)>0$, choose an $n_1>k$ such that $\mu (A \cap T^{-n_1} B) = \mu(A)\mu(B)$, and let $A_1:=A \cap T^{-n_1} B$.
Inductively, if $A_{m-1}\subset A$ with $\mu(A_{m-1})=\mu(A)\mu(B)^{m-1}$ has already been defined, use the hypothesis to find $n_{m}>k$ such that $A_{m}:=A_{m-1} \cap T^{-n_{m}} B$ satisfies $$\mu(A_{m})=\mu(A_{m-1})\mu(B) = \mu(A)\mu(B)^{m} \,.$$
Given $\epsilon>0$, choose $m$ such that $\mu(A_{m})=\mu(A)\mu(B)^{m}<\epsilon$. Then $d(A,A\setminus A_m) <\epsilon$, and $$\mu((A\setminus A_m) \cap T^{-n_{m}} B)= \mu( A \cap T^{-n_{m}} B)-\mu(A_m)$$ $$=\mu(A)\mu(B)-\mu(A_m) \ne \mu(A\setminus A_m)\mu(B)\,.$$ Thus $A\setminus A_m \in H_k$, and for the same reason, $A_m\in H_k$. To approximate the equivalence class of the empty set, use the above construction with $\Omega$ in place of $A$, to obtain a measurable set $\Omega_m \in H_k$ with $\mu(\Omega_m)=\mu(B)^{m}<\epsilon$. Thus $d(\Omega_m,\emptyset)<\epsilon$, and we have verified that $H_k$ is dense in $(M,d)$.
By the Baire-Category theorem, $\cap_{k \ge 1}H_k$ is nonempty, and any set $A$ in this intersection violates the hypothesis.