For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself.

For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define $h(\mathcal{A},T,m)$ as $\lim_{n\rightarrow \infty}\frac{1}{n}H(\bigvee_{i=0}^{n-1}T^{-i}\mathcal{A})$. and measurable h(T,m) is to take sup of all possible finite measurable partition.

It is subtle as least to me to understand $h(T^{n},m)=n h(T,m) (n>0)$. Of course I can prove it almost by definition. However I can not feel the essence of measurable entropy.

For topological entropy, we can understood $h(T^n)=nh(T)$ very well using Bowen balls related to numbers of equivalence of (cut-off) orbits.

I wondered whether there exist similar explanation for measurable entropy. Any reference and commnents will be greatly appreciated.