Proposition 8

— For every pair of numerical functions $f \geq 0$, $f' \geq 0$ defined on $T$ and $T'$, *respectively,

We begin by treating the case that $\mu$ and $\mu'$ are measures with compact support; the same is then true of $\mu \otimes \mu'$, and all of the symbols $\int^\bullet$, $\iint^\bullet$ may be replaced by upper integrals. By Prop. 6,

To establish the reverse inequality, let us choose a function $h \geq f$ (resp. $h' \geq f'$), the lower envelope of a sequence $(h_n)$ (resp. $(h'_n)$) of lower semi-continuous functions, such that

(resp. $\int^* h'(t')\, d\mu'(t') = \int^* f'(t')\, d\mu'(t')$); the existence of such functions follows immediately from the definition of upper integral (Ch. IV, §1, No. 3, Def. 3) and Lebesgue’s theorem. Applying Prop. 7 to the measurable function $h \otimes h'$, we have

which is the sought-for inequality. Thus the proposition is established when $\mu$ and $\mu'$ are measures with compact support. To treat the general case, it suffices to represent $\mu$ (resp. $\mu'$) as the sum of a family $(\mu_\alpha)_{\alpha \in A}$ (resp. $(\mu'_\beta)_{\beta \in B}$) of measures with compact support (§ 2, No. 3, Prop. 4), write the formula (10) for each measure $\mu_\alpha \otimes \mu'_\beta$, and sum on $(\alpha, \beta)$, taking into account Prop. 1 (§ 2, No. 2, Prop. 1).