Let (X, S, μ) and (Y, T, ν) be two measure spaces,K(g)=∫ Y k(x,y)g(y)dv(y) ξ +=max (ξ,0), and $$\delta (K) = \sup _{x_1 ,x_2 \in X} \int {{}_Y(k(x_1 } y) - (k(x_2 ,y))^ + dv(y)$$ . Two integral inequalities (the first of which has a simple geometrical interpretation) involvingδ(K) are proved. Generalizations of theorems about infinite stochastic matrices and convergence of superpositions of sequences of integral operators are obtained.