We investigate the best approximations <$E e sub sigma (f)> of the integrals of functions of the spaces <$E L sub p (A, d mu )> by the integrals of rank <$E sigma>. We find the exact values of the upper bounds of these approximations in the case where the function f is a product of two functions, one of which is fixed and the other one changes in some subset of <$E L sub p (A, d mu )>. We also obtain, in terms of approximations <$E e sub sigma ( cdot )>, the necessary and sufficient conditions for a function of the space <$E L sub p (A, d mu )> to belong to the space <$E L sub s (A, d mu )>, <$E 0~<<~p,~s~<<~inf>.