Analysis of Parametrized Measures

Abstract

In this chapter we shall analyze more closely parametrized measures and introduce the basic tools to deal with these families of probability measures. Some of these will be used several times later. Our main goal here is to characterize parametrized measures: we are interested in knowing when a given family of probability measures can actually be generated as the parametrized measure by some sequence of functions. At this stage we do not place any further restriction on the sequences we would like to consider except for boundedness in some L^pΩ.In this regard we place ourselves in the context of Section 2 of Chapter 2. As a matter of fact, the main theorem of this chapter, Theorem 7.7, can be proved directly taking advantage of the analysis carried out there and extending it to the casepfinite by means of some technicalities involving truncation operators. This will actually be our approach to pass from p = ∞to finitepin Chapter 8 under the gradient constraint. Nonetheless we have chosen to proceed in a different way with the idea in mind of preparing some of the main techniques for the analysis of gradient parametrized measures pursued in Chapter 8.