Exercise 7, § 8

a) Let T, T' be two locally compact spaces; assume that T' admits a countable base. Let $\mu$ be a positive measure on T, $\mu'$ a positive measure on T'. Let $f$ be a numerical function $\geq 0$ defined on $T \times T'$, bounded on every compact subset of $T \times T'$, and such that: $1^\circ$ for almost every $t \in T$, the function $t' \mapsto f(t, t')$ is $\mu'$-measurable; $2^\circ$ for every function $h \in \mathscr{K}(T')$, the function

defined almost everywhere, is $\mu$-measurable. Show that, under these conditions, there exists a function $g$, measurable for $\mu \otimes \mu'$, such that for every $t \in T$, one has $f(t, t') = g(t, t')$ except at the points of a $\mu'$-negligible set $A_t$ (depending on $t$). (Show that for every function $\varphi \in \mathscr{K}(T \times T')$, the function $t' \mapsto f(t, t') \varphi(t, t')$ is $\mu'$-integrable for almost every $t \in T$, and that the function $t \mapsto \int f(t, t') \varphi(t, t') d\mu'(t')$, defined almost everywhere, is $\mu$-integrable; one makes use of Lemma 1 of Ch. III, §4, No. 1. Observe next that $\varphi \mapsto \int d\mu(t) \int f(t, t') \varphi(t, t') d\mu'(t')$ is a positive measure on $T \times T'$, with base $\mu \otimes \mu'$, and apply the Lebesgue–Nikodym theorem; finally, use the fact that there exists a countable dense set in $\mathscr{K}(T')$.)

b) Suppose, moreover, that $T$ is metrizable. Show that the conditions of a) are satisfied if: $1^\circ$ for almost every $t' \in T'$, the function $t \mapsto f(t, t')$ is $\mu$-measurable; $2^\circ$ for almost every $t \in T$, the function $t' \mapsto f(t, t')$ is continuous almost everywhere (for $\mu'$). (Use Exer. 13 of Ch. IV, §5.)

c) Take $T = T' = [0, 1]$. Admitting the continuum hypothesis (S, Ch. III, §6, No. 4), let $x \prec y$ be a well-ordering relation on $T$ for which there is no greatest element, and such that for every $x \in T$, the set of $z \prec x$ is countable. If one takes for $\mu = \mu'$ Lebesgue measure, show that the characteristic function $f$ of the set of pairs $(t, t')$ such that $t \prec t'$ satisfies the conditions of a), but that $\int f(t, t') d\mu(t) = 0$ for every $t' \in T'$, and $\int f(t, t') d\mu'(t') = 1$ for every $t \in T$.