Exercise 2, § 5

Let $g$ be a positive function on T, measurable and essentially bounded for $\mu$, and let $\nu = g \cdot \mu$. Show that if a function f defined on T, with values in a Banach space or in $\overline{\mathbf{R}}$, is $\mu$-integrable, then f is $\nu$-integrable. (Observe that $\nu \leq a \mu$ for some constant $a > 0$, and make use of Prop. 15 of Ch. IV, §1, No. 3.)