Exercise 7, § 2

Let $\mu$ be a positive measure on T, m a measure on T with values in F, having base $\mu$ and density f with respect to $\mu$. Let q be a lower semi-continuous semi-norm on F. Show that if, for every compact subset K of T, $\int_K^*(q \circ f)\, d\mu$ is finite, then, for every $\mu$-measurable function $h \geq 0$, the inequality $\int^* h\, dq(m) \leq \int^*(q \circ f)h\, d\mu$ holds. (Use Lemma 1 of Ch. V, 1st edn., §2, No. 2.)²