Remark

Lemmas 1,2,3 can be proved very simply without making use of the results of §§4 and 6, by a direct argument. For example, the relation (2) is obvious by definition if $f \in \mathscr{K}(T \times T')$. If $f$ is lower semi-continuous on $X = T \times T'$, it suffices to observe that $t' \mapsto f_t(t')$ is the upper envelope of the functions $t' \mapsto g_t(t') = g(t, t')$, where $g$ runs over the set of functions in $\mathscr{K}(X)$ such that $0 \leq g \leq f$. Finally, for arbitrary $f$, one notes that if $h \geq f$ is lower semi-continuous on $X$, then $t' \mapsto h(t, t')$ is lower semi-continuous on $T'$; and conversely, if $t' \mapsto u(t')$ is lower semi-continuous on $T'$ and is such that $u(t') \geq f(t, t')$ for all $t' \in T'$, then the function $h$ such that $h(t, t') = u(t')$, $h(t_1, t') = +\infty$ for $t_1 \neq t'$, is lower semi-continuous on $X$ and satisfies $h \geq f$. Once Lemma 1 is proved, one deduces from it that the set $(T - \{t\}) \times T'$ is $\lambda'_t$-negligible, and it is then every easy to prove Lemmas 2 and 3.

The relation (3) permits denoting its two members by $\int f(t, t') \, d\mu'(t')$ without risk of confusion. The analogous results obviously hold for the measures $\lambda_{t'} = \mu \otimes \varepsilon_{t'}$.

Instead of the notations

we shall employ the notations

consistent with the notations adopted in Ch. III, §4, No. 1.

The interpretation of the measure $\nu$ as an integral $\int \lambda'_t \, d\mu(t)$ will allow us to translate the results of §3 into the language of product measures. On the other hand, the measure $\lambda'_t$ is carried by $\{t\} \times T' = \overline{\mathrm{pr}}_1^{-1}(t)$, so that this integral defines a decomposition of $\nu$ into slices, relative to the projection $\mathrm{pr}_1$ of $T \times T'$ onto $T$ (§6, No. 6). Before giving a list of the results so obtained, here is a useful property: