= 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 $t\geq0$,
$$
\mu(A\cap\{F>t\})\leq e^{-t}.
$$
Together with the bound by $\mu(A)$ and the <tail integral formula for moments>, this yields $\int F\,d\mu\leq H_\mu(\xi)+1$. The inequality follows by stopping the <conditional-expectation martingale> $\mathbb E[\mathbf1_A\mid\mathcal G_n]$ the first time it falls below $e^{-t}$.
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