Bourbaki

Corollary 1

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

Integration, chapter V, § 8. Products of measures

— Let ff (resp. f′f') be a lower semi-continuous function ⩾0\geqslant 0 defined on TT (resp. T′T'); the function f⊗f′:(t,t′)↦f(t)f′(t′)f \otimes f' : (t, t') \mapsto f(t)f'(t') is then lower semi-continuous on T×T′T \times T', and

∬∗f(t)f′(t′) dμ(t) dμ′(t′)=(∫∗f(t) dμ(t))(∫∗f′(t′) dμ′(t′)). \iint^* f(t)f'(t') \, d\mu(t) \, d\mu'(t') = \left( \int^* f(t) \, d\mu(t) \right) \left( \int^* f'(t') \, d\mu'(t') \right).

Let GG (resp. G′G') be the set of functions g∈K+(T)g \in \mathscr{K}_+(T) (resp. g′∈K+(T′)g' \in \mathscr{K}_+(T')) such that g⩽fg \leqslant f (resp. g′⩽f′g' \leqslant f'); then

f⊗f′=sup⁡g∈G, g′∈G′g⊗g′. f \otimes f' = \sup_{g \in G,\ g' \in G'} g \otimes g'.

Since the functions g⊗g′g \otimes g' belong to K+(T×T′)\mathscr{K}_+(T \times T'), f⊗f′f \otimes f' is indeed lower semi-continuous, and (6) follows at once from Prop. 5 (or directly by passage to the limit in the preceding formula).

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

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Transcribed from the printed volume. Not translated.