  We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\}$ on a
metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the
conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta
\Big) =1 when one has the \textit{a priori} estimate on the modulus of
continuity and the value at one point. We compare our problem to the celebrated
Kolmogorov continuity criteria for stochastic processes, and finally give an
application of our main result for stochastic intergrals with respect to
compound Poisson random measures with infinite intensity measures.
