Theorem 2 (Hölder’s inequality)

— Let $f$ and $g$ be two numerical functions that are finite almost everywhere and are such that $f$ is equal almost everywhere to a function in $L^p$ and $g$ to a function in $L^q$. Then, the function $fg$ (defined almost everywhere) is integrable, and

Let $f_1$ (resp. $g_1$) be a function in $L^p$ (resp. $L^q$) to which $f$ (resp. $g$) is equal almost everywhere; $fg$ is equal almost everywhere to the function $f_1 g_1$, which is everywhere defined and finite, and which is measurable, being the product of two measurable functions (§ 5, No. 3, Cor. 5 of Th. 1). If $1 < p < +\infty$, Hölder’s inequality for the upper integral (Ch. I, No. 3, Prop. 4) yields the inequality (4), and the relation $N_1(fg) < +\infty$ then shows that $fg$ is integrable (§ 5, No. 6, Th. 5). If $p = 1$, $q = +\infty$, the inequality (4) and the fact that $fg$ is integrable are immediate consequences of the inequality of the mean (No. 2, Prop. 1); thus the theorem is proved in all cases.