Differentiation and Duality

The work on measures is extended to scalar set functions in order to analyze their variational properties. The chapter establishes the Hahn and Lebesgue decompositions. After considering concretely the absolutely continuous and completely monotone functions on the line, it presents the Radon-Nikodym theorem along with its general forms, and related differentiation results. As applications, the chapter gives a brief account of likelihood ratios of finite measures and the duality of L^p -spaces, together with essential properties of conditional expectations; and includes a few results on martingales. It considers the comparison of a pair of additive set functions and then specialize the property to point functions. This gives a feeling for the concept, its deeper implications, and a motivation for the general case. The chapter also considers the general form to obtain optimum conditions for the validity of the Radon-Nikodym theorem.