Proposition 6

— Let $X$ be a discrete space, $\mu$ the measure on $X$ defined by placing a mass $+1$ at each point of $X$. If $f$ is a mapping of $X$ into the 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 right end-point $+\infty$ and containing this point; moreover, $N_p(f)$ is a decreasing function of $p$ on I.

For, $\mu^*(|f|) = \sum_{x \in X} |f(x)|$ for every function $f$ (§ 1, No. 1, Example), and $N_\infty(f) = \|f\| = \sup_{x \in X} |f(x)|$; if there exists a number $\alpha > 0$ such that $|f(x)| \geq \alpha$ for infinitely many values of $x \in X$, then $N_p(f) = +\infty$ for every finite $p$; in the contrary case, there exists an $x_0 \in X$ such that $|f(x_0)| = \|f\|$, whence

for every finite $p$. Since the function $\log N_p(f)$ is convex with respect to $1/p$ and takes its smallest value at the point $+\infty$, it is necessarily a decreasing function of $p$ on I (FRV, I, § 4, No. 3, Prop. 5), which completes the proof.