A Characterization of Vitali Conditions in Terms of Maximal Inequalities

Abstract

Vitali conditions $V, V', V_p, 1 \leqslant p < \infty$, on $\sigma$-algebras indexed by a directed set, are shown to hold if and only if the maximal inequality \begin{equation*}\tag{1} P(\text{essential} \lim \sup X_t \geqslant \alpha) \leqslant K \lim \sup_{T^\ast}E(X_\tau)/\alpha\end{equation*} holds for all adapted positive processes $(X_t)$, and all positive numbers $\alpha$. Here $K$ is a constant which may be taken equal to 1, and $T^\ast$ is the appropriate directed set of stopping times: for $V, T^\ast$ is the set of simple stopping times; for $V', T^\ast$ is the set of simple ordered stopping times; for $V_p, T^\ast$ is the set of multivalued stopping times with overlap going to zero in $L_p$. The inequality (1) is true whatever be the $\sigma$-algebras, provided that essential $\lim \sup$ is replaced by stochastic $\lim \sup$.