Chapter VII

Haar measure

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 $\mathscr{K}(T)$ will denote either the space $\mathscr{K}_\mathbf{R}(T)$ or the space $\mathscr{K}_\mathbf{C}(T)$; similarly for the notations $\overline{\mathscr{K}(T)}$, $\mathscr{C}(T)$, $L^p(T, \mu)$, $\mathscr{M}(T)$, 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 $\overline{\mathscr{K}(T)}$ will always be assumed to be equipped with the topology of uniform convergence, the space $\mathscr{C}(T)$ with the topology of compact convergence, and the space $\mathscr{K}(T)$ with the direct limit topology whose definition is reviewed at the beginning of Chapter VI. The notation $\mathscr{K}_+(T)$ will denote the set of functions $\geqslant 0$ of $\mathscr{K}(T)$. If $A \subset T$, $\varphi_A$ will always denote the characteristic function of $A$. If $t \in T$, $\varepsilon_t$ will denote the positive measure defined by the mass +1 at the point $t$.

All locally convex spaces will be assumed to be Hausdorff.

We will denote by e the neutral element of all of the groups considered, absent express mention to the contrary.