Power-Integrable Functions and the p-Seminorm

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\ge1$ be a real number. For real $t\ge0$ and $a>0$, $t^{a}$ is the real power of Real Powers (i) and Real Powers (ii).

(i) The function $|f|^{p}$. For a measurable $f:X\to\mathbb{R}$, $|f|^{p}$ is the function

$$|f|^{p}:X\to\mathbb{R},\qquad x\mapsto|f(x)|^{p}$$

(an element of $\mathcal{M}^{+}$ by Real Powers of the Absolute Value of a Measurable Function Are Measurable (i)).

(ii) Power-integrable functions. $\mathcal{L}^{p}(\mu)=\mathcal{L}^{p}(X,\mathcal{F},\mu)$ is the set of measurable functions $f:X\to\mathbb{R}$ with

$$\int|f|^{p}\,d\mu<\infty,$$

the integral of $|f|^{p}$ as an element of $\mathcal{M}^{+}$. Its elements are the $p$-integrable functions on $(X,\mathcal{F},\mu)$.

(iii) The $p$-seminorm. For $f\in\mathcal{L}^{p}(\mu)$, $\int|f|^{p}\,d\mu$ is a real number $\ge0$ (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii) and The Extended Real Line, Its Order, and [0,∞] (ii)), and the $p$-seminorm of $f$ is

$$\|f\|_{p}=\Bigl(\int|f|^{p}\,d\mu\Bigr)^{1/p}.$$