Proposition 5

— Let $\mu$ be a bounded measure, and let $f$ be a $\mu$-measurable function with values in a Banach space $F$. The set I of numbers $p$ such that $1 \leq p \leq +\infty$ and $N_p(f)$ is finite, is either empty or is an interval with left end-point $p = 1$ and containing this point; moreover, $(|\mu|(X))^{-1/p} N_p(f)$ is an increasing function of $p$ on I.

This is an immediate consequence of Prop. 4 above and of the Cor. of Prop. 4 of Ch. I, No. 3.