Measures

This chapter defines the space of measures on an open subset, with values in a Neumann space. Since integrable measures will be defined as limits of sequences of continuous functions, these functions are identified with measures. Indeed, every continuous function with compact support is uniformly continuous, according to a corollary of Heine's theorem. The chapter shows that two distinct continuous functions yield distinct measures, which will identify each continuous function with a measure. It also defines the regularizing sequences and the localizing sequences used to construct them, and the regularizations of a measure. The chapter focuses on the construction of the regularizations of a measure, which we will verify to be continuous approximations with compact support. They will serve to characterize integrable measures and to establish many of their properties. The chapter states a continuity property of the regularization operators. It presents some properties of the regularizations of a continuous function.