Proposition 4

Tag 09N9, permanent. Cite it as [Bourbaki, Tag 09N9].

Integration, chapter IX, § 1. Premeasures and measures on a topological space

— Let $f$ be a positive function and $(g_n)_{n \geq 1}$ a sequence of $\mu$ -measurable positive functions on T. Setting $g = \sum_{n \geq 1} g_n$ , one has

\mu^\bullet(fg) = \sum_{n \geq 1} \mu^\bullet(fg_n).

Set $h_n = \sum_{i=1}^n g_i$ for all $n \geq 1$ . For every compact subset K of T,

\mu_K^\bullet((fh_n)_K) = \sum_{i=1}^n \mu_K^\bullet((f g_i)_K)

by Prop. 2 of Ch. V, §1, No. 1 applied to the compact space K. Passing to the limit with respect to the increasing directed set of compact subsets of T, one obtains

\mu^\bullet(fh_n) = \sum_{i=1}^n \mu^\bullet(f g_i).

Now, $fg$ is the limit of the increasing sequence $(fh_n)_{n \geq 1}$ , whence $\mu^\bullet(fg) = \lim_{n \to \infty} \mu^\bullet(fh_n)$ ; the preceding formula then immediately implies (4).

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

Proposition 4

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