Let T be a measure preserving transformation on the probability space $(X,\mathcal{F},\mu)$.
I have already solved this problem:
Suppose $\alpha$ is a finite partition of $X$. Show that $h_{\mu}(\alpha, T) = h_{\mu}(\bigvee_{i=1}^n T^{-i}\alpha,T)$ for any $n\in\mathbb{N}$.
But after a day I could not find a solution to the next one:
Let $\alpha$ and $\beta$ be finite partitions. Show that $h_{\mu}(\beta,T)\leq h_{\mu}(\alpha,T)+H_{\mu}(\beta|\alpha)$
I think it is undoable to do it by using the definition of entropies. Since $h_{\mu}(\alpha,T)\leq H_{\mu}(\alpha)$ and $H_{\mu}(\beta|\alpha) = H_{\mu}(\alpha\vee\beta)-H_{\mu}(\alpha)$, it suffices to show
$H_{\mu}(\beta)+H_{\mu}(\alpha)\leq h_{\mu}(\alpha,T)+H_{\mu}(\alpha\vee\beta)$
Maybe the previous problem should be used, but the next problem (c) says "use parts a and b to show...", so it is not very likely.
Is there anyone who can help? Thanks in advance.