Power-Integrable Functions and the p-Seminorm

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p$ be a real number with $1\le p$.

1. (The power of the absolute value is measurable)¶ For a measurable $f:X\to\mathbb{R}$ let $|f|^{p}$ denote the map $X\to\mathbb{R}_{+}$ sending $x$ to the power $(|f(x)|)^{p}$. This map is measurable. Indeed $|f|$ is measurable by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and takes nonnegative values; for a real $c<0$ the set $\{x\in X:c<(|f(x)|)^{p}\}$ is $X$, which lies in $\mathcal{F}$, and for a real $c$ with $0\le c$ the comparison in Properties of Real Powers of Nonnegative Real Numbers §inverse gives $c<(|f(x)|)^{p}$ exactly when $c^{1/p}<|f(x)|$, so that

Measurability now follows from the criterion recorded in Measure Spaces and the Lebesgue Integral: Standing Notation §measurable.

2. (Power-integrable functions)¶ The set

is the set of $p$-integrable functions on $(X,\mathcal{F},\mu)$; the integral is that of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, formed for the measurable map of clause 1, and is a member of $[0,\infty]$. Members of $\mathcal{L}^{p}(X,\mathcal{F},\mu)$ are regarded as elements of the real vector space of all real-valued maps on $X$, with the pointwise sum and pointwise scalar multiple defined there.

3. (The $p$-seminorm)¶ For $f\in\mathcal{L}^{p}(X,\mathcal{F},\mu)$ the number $\int_{X}|f|^{p}\,d\mu$ is a nonnegative real number, so its power with the positive exponent $1/p$ is defined. The $p$-seminorm of $f$ is

The notation suppresses the measure space, which is always the one fixed above.