Remark

— Let $(K_\alpha)_{\alpha \in A}$ be a crushing of $S$ for $\lambda$ and let $M = S - \bigcup_{\alpha \in A} K_\alpha$; define in an analogous way $(L_\beta)_{\beta \in B}$ and $N$ for the measure $\mu$ on $T$. We denote by $S'$ the locally compact space that is the sum of the subspaces $K_\alpha$ of $S$ and the discrete space $M$; the space $T'$ is defined analogously, and we set $X' = S' \times T'$. The locally compact space $X'$ is the sum of the family $(K_\alpha \times L_\beta)_{(\alpha, \beta) \in A \times B}$ of compact subspaces of $X$ and the subspace $P = (M \times T) \cup (S \times N)$ that is a locally $\nu$-negligible subset of $X$ (one observes that $P$ is in general not a discrete space). We saw in the Scholium of $§ 1$, No. 8 that there exists a measure $\lambda'$ on $S'$ such that the measurable functions, the essential upper integral of positive functions, the essentially integrable functions and their integrals, are the same for $\lambda$ and $\lambda'$. We associate the measure $\mu'$ on $T'$ with $\mu$, and the measure $\nu'$ on $X'$ with $\nu$, in conformity with the cited Scholium; one sees immediately that ${\nu'}^\bullet(f \otimes g) = {\lambda'}^\bullet(f) {\mu'}^\bullet(g)$ for $f \in \mathscr{F}_+(S)$ and $g \in \mathscr{F}_+(T)$; therefore $\nu' = \lambda' \otimes \mu'$ by Prop. 10 of No. 5. Since the topology of $X'$ is finer than that of $X$, every $\nu$-moderated function is $\nu'$-moderated. This procedure permits extending without new proof the Lebesgue–Fubini theorem (Ch. V, $§ 8$, No. 4, Th. 1) to the present situation.