Definition 1

Tag 09PR, permanent. Cite it as [Bourbaki, Tag 09PR].

Integration, IX, p. 44, § 3. Measures and additive set functions

Let $T$ be a topological space, and let $\mathscr{B}(T)$ be the Borel tribe of $T$ ; let $I$ be a mapping of $\mathscr{B}(T)$ into $\overline{\mathbf{R}}_+$ . a) $I$ is said to be countably additive if, for every sequence $(A_n)$ of pairwise disjoint elements of $\mathscr{B}(T)$ ,

I\left( \bigcup_n A_n \right) = \sum_n I(A_n).

I

is said to be inner regular if, for every set

A \in \mathscr{B}(T)

I(A) = \sup_K I(K),

where

K

runs over the set of compact subsets of

A

c) I is said to be bounded (resp. locally bounded) if $I(T) < +\infty$ (resp. if every point $x \in T$ admits an open neighborhood $V$ such that $I(V) < +\infty$ ).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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