Chapter IX

Measures on Hausdorff topological spaces

If T is a set, and A is a subset of T, we denote by $\varphi_A$ the characteristic function of A, provided this does not lead to any confusion. The set $\overline{\mathbf{R}}_+^T$ of numerical functions $\geq 0$ (finite or not) defined on T will be denoted by $\mathscr{F}_+(T)$, or simply $\mathscr{F}_+$ if there is no ambiguity as to T; this set will always be equipped with its natural order structure. Recall that the product of two elements of $\mathscr{F}_+$ is always defined, thanks to the convention $0 \cdot (+\infty) = 0$. If A is a subset of T, and f is a function defined on T, the restriction $f|A$ of f to A may be denoted $f_A$ in this chapter, if this creates no confusion; an analogous notation will be employed for induced measures. On the other hand, if $f \in \mathscr{F}_+(A)$ we shall denote by $f^0$ the extension by 0 of f to T, that is, the function defined on T that coincides with f on A and with 0 on $T - A$.

All topological spaces considered in this chapter are assumed to be Hausdorff, absent express mention to the contrary. From §1, No. 4 on, except for §5, all measures will be assumed to be positive, absent express mention to the contrary.