Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function $g(x)$, in order that $Mg(\xi_1+\xi_2+\dots+\xi_\nu)<\infty$, if $Mg(\xi_i)<\infty$ and $Mg(\nu)<\infty$, $\nu,\xi_1,\xi_2,\dots,\xi_n,\dots$ being non-negative and independent, $\nu$ being integral, and $\{\xi_i\}$ being identically distributed. The result is applied to the theory of branching processes.