Exercise 11, § 1

Let $\mu$ be Lebesgue measure on $T = [0,1]$, $F$ a Hilbert space having a countable orthonormal basis $(e_n)$. Let $f$ be the mapping of $T$ into $F$ such that $f(0) = 0$ and $f(t) = 2^n e_n / n$ on the interval $]2^{-n-1}, 2^{-n}]$ ($n \geq 0$). Show that $f$ is $\mu$-measurable, scalarly $\mu$-integrable, and that $\int f\,d\mu \in F$, but that $\int^* |f|\,d\mu = +\infty$.