  We take an $L_1$-dense class of functions $\Cal F$ on a measurable space
$(X,\Cal X)$ together with a sequence of independent, identically distributed
$X$-space valued random variables $\xi_1,\dots,\xi_n$ and give a good estimate
on the tail distribution of $\sup_{f\in\Cal F}\sum_{j=1}^n f(\xi_j)$ if the
expected values $E|f(\xi_1)|$ are very small for all $f\in\Cal F$. In a
subsequent paper~[2] we shall give a sharp bound for the supremum of normalized
sums of i.i.d. random variables in a more general case. But that estimate is a
consequence of the results in this work.
