Proposition 5

Tag 097B, permanent. Cite it as [Bourbaki, Tag 097B].

— Let $f$ be a numerical function $\geqslant 0$ , lower semi-continuous on $T \times T'$ . Then, the function

t \mapsto \int^* f(t, t') \, d\mu'(t')

is lower semi-continuous on $T$ , and

\iint^* f(t, t') \, d\mu(t) \, d\mu'(t') = \int^* d\mu(t) \int^* f(t, t') \, d\mu'(t').

This is a consequence of Prop. 2 of §3, No. 1, taking into account Lemma 1 of No. 1.

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

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