for $\sigma$-algebras $\A_1$, $\A_2$,…, $\A_1\vee \A_2\vee \cdots$ or $\bigvee_{i} \A_i$ denotes the $\sigma$-algebra generated by $\bigcup_i \A_i$.

In the following lemma, we list some basic properties of the (conditional) information and entropy.

\[lem-par\] Let $T$ be a measure-preserving transformation of a separable probability space $(X,\B,m)$. Let $\xi,\eta$ be two countable Borel partitions of $X$ with $H_m(\xi)<\infty$, $H_m(\eta)<\infty$, and $\A$ a sub-$\sigma$-algebra of $\B$. Then we have

-   ${\bf E}_m(f|\A)\circ T={\bf E}_m(f\circ T|T^{-1}\A)$ for $f\in L^1(X, \B, m)$.

-   ${\bf I}_{m}(\xi|\A)\circ T={\bf I}_m(T^{-1}\xi|T^{-1}\A)$.

-   ${\bf I}_m(\xi\vee \eta|\A)={\bf I}_m(\xi|\A)+{\bf I}_m(\eta|\widehat{\xi}\vee \A)$.

-   $H_m(\xi\vee \eta|\A)=H_m(\xi|\A)+H_m(\eta|\widehat{\xi}\vee \A)$.

-   If $\A_1\subset \A_2\subset\cdots $ is an increasing sequence of sub-$\sigma$-algebras with $\A_n\uparrow \A$, then $\sup_n{\bf I}_m(\xi|\A_n)\in L^1$, and ${\bf I}_m(\xi|\A_n)$ converges almost everywhere and in $L^1$ to ${\bf I}_m(\xi|\A)$. In particular, $\lim_{n\to \infty} H_m(\xi|\A_n)=H_m(\xi|\A)$.

Below we present an additional property of the conditional expectation.

\[lem-2.10\] Let $(X,\B,m)$ be a probability space and $\A$ a sub-$\sigma$-algebra of $\B$. Let $A\in \B$ with $m(A)>0$. Then $$\E_m(\chi_A|\A)(x)>0$$ for $m$-a.e. $x\in A$.

The following lemma is a variant of Maker’s ergodic theorem ([@Maker1940]).

\[lem-3.15\] Let $T$ be a measure-preserving transformation of a probability space $(X,\B,m)$. Let $g_k\in L^1(X, \B, m)$ be a sequence that converges almost everywhere and in $L^1$ 