Extremal problems for integrals of non-negative functions

We study the numbers $e_\sigma(f)$ that characterize the best approximation of the integrals of functions in $L_p(A,d\mu)$, $p>0$, by integrals of rank $\sigma$. We find exact values and orders as $\sigma\to\infty$ for the least upper bounds of these numbers on the classes of functions representable as products of a fixed non-negative function and functions in the unit ball $U_p(A)$ of $L_p(A,d\mu)$. The numbers $e_\sigma(\,\cdot\,)$ are used to obtain necessary and sufficient conditions for an arbitrary function in $L_p(A,d\mu)$ to lie in $L_s(A,d\mu)$, $0<p,s<\infty$. We discuss applications of the results obtained to the approximation of measurable functions (given by convolutions with summable kernels) by entire functions of exponential type.