Corollary 1

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

Let $f$ be a positive numerical function defined on $X = S \times T$ ; then

\iint_X^* f(s,t)\, d\nu(s,t) \geq \int_S^* d\lambda(s) \int_T^* f(s,t)\, d\mu(t).

For, let $g$ be a lower semi-continuous function on $X$ such that $g \geq f$ ; by Prop. 11,

\iint_X^* g(s,t)\, d\nu(s,t) = \iint^* g(s,t)\, d\nu(s,t) = \int^* d\lambda(s) \int^* g(s,t)\, d\mu(t)

= \int^* d\lambda(s) \int^* g(s,t)\, d\mu(t) \geq \int^* d\lambda(s) \int^* f(s,t)\, d\mu(t).

The inequality (10) is obtained by passing to the lower envelope over

g

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

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