Haar measure

Abstract

In this chapter and in the next, when we speak of a function (resp. a measure), it will be understood to be either a real or a complex function (resp. measure); if T is a locally compact space, the notation K(T) will denote either the space K_R(T) or the space K_C(T); similarly for the notations $\overline {\chi (T)}$, C(T), L^p(T,μ), M(T), etc. It is of course understood that in a situation involving several functions, measures or vector spaces, the results obtained are valid when these functions, measures or vector spaces are all real or all complex. The space $\overline {\chi (T)}$ will always be assumed to be equipped with the topology of uniform convergence, the space C(T) with the topology of compact convergence, and the space C(T) with the direct limit topology whose definition is reviewed at the beginning of Chapter VI. The notation C₊(T) will denote the set of functions ≥ 0 of C(T). If A ⊂ T, φ_A will always denote the characteristic function of A. If t Є T, ε_t will denote the positive measure defined by the mass +1 at the point t.