Integration of measures

Abstract

Throughout this chapter, T denotes a locally compact space, μ a positive measure on T. For every subset A of a set E, ϕ_A denotes the characteristic function of A (if no confusion can result thereby). By numerical function, we always mean a function taking its values in $\overline {\text{R}}$, thus possibly taking on the values +∞ and −∞. The set of positive numerical functions defined on E will be denoted ℱ₊(E), or simply by ℱ₊ no confusion can result We agree to define the products 0·(+∞) and 0·(−∞) by giving them the value 0; thus, if f is a numerical function defined on E, and A is a subset of E, fϕ_A denotes the function that coincides with f on A and is equal to 0 on CA. For every point a of a locally compact space, ε_a denotes the measure defined by placing a unit mass at the point a (Ch. III, §1, No. 3).