Elementary Properties of the p-Seminorm

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

1. (Recovering the integral)¶ For every $f\in\mathcal{L}^{p}$,

2. (Absolute homogeneity)¶ Let $f\in\mathcal{L}^{p}$ and let $c$ be a real number. Then $cf\in\mathcal{L}^{p}$ and $\lVert cf\rVert_{p}=|c|\,\lVert f\rVert_{p}$.

3. (Comparison)¶ Let $g\in\mathcal{L}^{p}$ and let $f:X\to\mathbb{R}$ be measurable with $|f(x)|\le|g(x)|$ for almost every $x\in X$. Then $f\in\mathcal{L}^{p}$ and $\lVert f\rVert_{p}\le\lVert g\rVert_{p}$.

4. (Almost-everywhere equality)¶ Let $f\in\mathcal{L}^{p}$ and let $g:X\to\mathbb{R}$ be measurable with $f(x)=g(x)$ for almost every $x\in X$. Then $g\in\mathcal{L}^{p}$ and $\lVert g\rVert_{p}=\lVert f\rVert_{p}$.

5. (Vanishing seminorm)¶ Let $f\in\mathcal{L}^{p}$. Then $\lVert f\rVert_{p}=0$ if and only if $f(x)=0$ for almost every $x\in X$.

6. (Rescaling by a power)¶ Let $r$ and $s$ be real numbers with $0<r$, $1\le s$ and $1\le rs$, let $h:X\to\mathbb{R}$ be measurable, and let $|h|^{r}$ denote the map $X\to\mathbb{R}_{+}$ sending $x$ to $(|h(x)|)^{r}$. Then $|h|^{r}$ is measurable, and $h\in\mathcal{L}^{rs}$ if and only if $|h|^{r}\in\mathcal{L}^{s}$; in that case