Hoelder's Inequality, for Two and for Finitely Many Factors

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. For a real number $r$ with $1\le r$ write $\mathcal{L}^{r}$ for the set of $r$-integrable functions on $(X,\mathcal{F},\mu)$ and $\lVert\cdot\rVert_{r}$ for the $r$-seminorm. Finite products of real numbers are the finite products of that definition. Then the following hold.

1. (Hölder's inequality)¶ Let $p,q$ be conjugate exponents, let $f\in\mathcal{L}^{p}$ and let $g\in\mathcal{L}^{q}$. Then the pointwise product $fg$ belongs to $\mathcal{L}^{1}$ and

2. (Finitely many factors)¶ Let $n\in\mathbb{N}$, and for each $k\in[n]$ let $p_{k}$ be a real number with $1\le p_{k}$ and let $f_{k}\in\mathcal{L}^{p_{k}}$. Suppose that

Let $F:X\to\mathbb{R}$ be the pointwise product of $f_{1},\dots,f_{n}$, given by $F(x)=\prod_{k=1}^{n}f_{k}(x)$. Then $F\in\mathcal{L}^{1}$ and