Exercise 11, § 5

Let $f$ be a measurable mapping of $X$ into a Banach space $F$. For every rational integer $n$ (positive or negative) let $A_n$ be the set of $x \in X$ such that $2^n \geqslant |f(x)| > 2^{n-1}$. For $f$ to be integrable, it is necessary and sufficient that the series with general term $2^n \mu(A_n)$ ($n \in \mathbf{Z}$) be convergent.