Convergence of entropy of converging partitions

Question

I think the following should hold, but I don't see a short proof and don't have a reference yet:

Let $(\Omega, B, \mu, \varphi)$ be a measure-preserving dynamical system. For a finite partition $Q = \{Q_{0},...,Q_{k-1}\}$ let $h_{Q}(\mu)$ the entropy of $\mu$ for $Q$, that is

$h_{\mu}(Q) = \lim_{n \to \infty}-\frac{1}{n}\sum \mu(Q_{i_{1}}\cap\varphi^{-1}(Q_{i_{2}})\cap...\cap\varphi^{-(n-1)}(Q_{i_{n}}))\log\mu(Q_{i_{1}}\cap\varphi^{-1}(Q_{i_{2}})\cap...\cap\varphi^{-(n-1)}(Q_{i_{n}})) $

where the sum is taken over all $0 \le i_{1},...,i_{n} \le k-1$. The assertion now is, that if $Q^{m} = \{Q^{m}_{0},...,Q^{m}_{k-1}\}$ is a sequence of partitions such that $\max \mu (Q^{m}_{i} \Delta Q_{i}) \to 0$ for $m \to \infty$ for a Partition $Q = \{Q_{0},...,Q_{k-1}\}$, than we have also $h_{Q^{m}}(\mu) \to h_{Q}(\mu)$ for $m \to \infty$.

Do you see any short proof or do you know a reference for a proof?

Answer 1

This property holds but the proof depends on what you already know or can use.

The best is to use Rohklin’s distance $$ d(P,Q) = H_\mu (P|Q)+H_\mu (Q|P) $$ (but if you don't know that it is, not really immediate to prove). Indeed, we know that $$ |h_\mu (\varphi,P)−h_\mu (\varphi,Q)| \le d(P,Q) $$ or, in your notation, $$ |h_P (\mu)−h_Q (\mu)| \le d(P,Q). $$

Recall that $$ H_\mu(P|Q)=-\sum_{C\in P,D\in Q}\mu(C\cap D)\log\frac{\mu(C\cap D)}{\mu(D)}. $$