Exercise 1, § 1

Consider the locally compact spaces T, T' identical to the interval [0, 1], and the measures $\mu, \mu'$ on T and T', respectively, identical to the Lebesgue measure. Let F be a Hilbert space admitting a countable orthonormal basis, arranged in a double sequence $(e_{mn})$; let $F' = F$ be its dual. Set $u_m(t) = e_{mn}$ for $(n-1)2^{-m} \leq t < n2^{-m}$ and $1 \leq n \leq 2^m$; set $f(t, t') = 2^m u_m(t)$ for $2^{-m} < t' \leq 2^{-m+1}$, and finally $f(t, 0) = 0$ and $f(1, t') = 0$. Show that $f$ is scalarly integrable for the product measure $\mu \otimes \mu'$, but that the function $t' \mapsto f(t, t')$ is not scalarly $\mu'$-integrable for any $t \in T$.