The primary aim of this chapter is to discuss the connection between the Hermite–Hadamard double inequality and Choquet’s theory. Noticed first by C. P. Niculescu [420], [421] (during the conference Inequalities 2001, in Timişoara), this connection led him to a partial extension of the majorization theory beyond the classical case of probability measures, using the so-called Steffensen–Popoviciu measures. Their main feature is to offer a large framework under which the Jensen–Steffensen inequality remains available. As a consequence, one obtains the extension of the left-hand side of Hermite–Hadamard inequality to a context involving signed Borel measures on arbitrary compact convex sets. A similar extension of the right-hand side of this inequality is known only in dimension 1, the higher dimensional case being still open.