On Paszkiewicz-type criterion for a.e. continuity of processes in $L^p$-spaces
In this paper we consider processes $X_t$ with values in $L^p$, $p\geq 1$ on subsets $T$ of a unit cube in $\mathbb R^n$ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing $f:\mathbb R_+\rightarrow\mathbb R_+$ $$ \|X_t-X_s\|_p\leq f(\|t-s\|),\quad s,t\in T. $$ We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set $T$. This criterion turns out to be necessary for a wide class of functions $f$. We use a geometrical Paszkiewicz-type characteristic of the set $T$. Our result generalizes in some way the classical theorem by Kolmogorov.