Exercise 4, § 8

a) Let T, T' be two locally compact spaces countable at infinity, $\mu$ a positive measure on T, $\mu'$ a positive measure on T'. Show that for every $(\mu \otimes \mu')$-measurable mapping $f$ of $T \times T'$ into $\mathbf{R}$, the function $F : t \mapsto \int^* |f(t,t')| d\mu'(t')$ is $\mu$-measurable (consider T' as the union of an increasing sequence of compact sets).

b) Suppose, moreover, that for almost every $t \in T$ (for $\mu$) the function $f(t,\cdot)$ is $\mu'$-integrable and that for almost every $t' \in T'$ (for $\mu'$) the function $f(\cdot,t')$ is $\mu$-integrable; finally, assume that the function (defined almost everywhere for $\mu$) $t \mapsto \int f(t, t')\ d\mu'(t')$ is $\mu$-integrable. Show that the function (defined almost everywhere for $\mu'$) $t' \mapsto \int f(t, t')\ d\mu(t)$ is then $\mu'$-integrable and

(By virtue of a), there is a partition of T formed by a $\mu$-negligible set N and a sequence $(K_n)$ of compact sets such that each of the functions $f \varphi_{K_n \times T'}$ is $(\mu \otimes \mu')$-integrable. Set

and apply Exer. 20 of §5.)

c) Let T be the interval $[0, 1]$ of $\mathbf{R}$, $\mu$ Lebesgue measure on T. For every integer $n > 0$, let $A'_n = \left[ \frac{1}{2n}, \frac{3}{2n+1} \right]$, $A''_n = \left[ \frac{3}{2n+1}, \frac{1}{2n-1} \right]$, and, in the product space $T \times T$, let $B'_n = A'_n \times A'_n$, $B''_n = A''_n \times A''_n$, $C'_n = A'_n \times A''_n$, $C''_n = A''_n \times A'_n$. Set $f(t, t') = 4^{n+1}$ in $B'_n$ and $B''_n$, $f(t, t') = -4^{n+1}$ in $C'_n$ and $C''_n$, for every integer $n > 0$, and $f(t, t') = 0$ at the other points of $T \times T$. Show that the two integrals $\int d\mu(t') \int f(t, t')\ d\mu(t)$ and $\int d\mu(t) \int f(t, t')\ d\mu(t')$ are defined and equal, but the function $f$ is not integrable for the measure $\mu \otimes \mu$.