In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. To apply our results, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities. Then we deal with the nonnegativity of some integrals with nonnegative convex functions. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals. Finally, we give necessary and sufficient conditions for the satisfaction of the integral Jensen inequality and the integral Lah–Ribarič inequality for signed measures. All the considered problems are also studied for special classes of convex functions. To prove the main assertions some approximation results for nonnegative convex functions are also developed.