Maximal inequality for conditional information functions

For increasing sigma-algebras ${\mathcal{G}}_n$, put $f_n=I_{\mu }(\xi \mid {\mathcal{G}}_n)$ and $F={\sup }_n{f}_n$. For each atom $A\in \xi$ and $t\ge 0$,Together with the bound by $\mu (A)$ and the tail integral formula for moments, this yields $\int Fd\mu \le H_{\mu }(\xi )+1$. The inequality follows by stopping the conditional-expectation martingale $\mathbb{E}[1_A\mid {\mathcal{G}}_n]$ the first time it falls below $e^{-t}$.