Elementary Properties of 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 measurable $f:X\to\mathbb{R}$, $|f|^{p}$ is the function of Power-Integrable Functions and the p-Seminorm (i); $\mathcal{L}^{p}(\mu)$ is the set of $p$-integrable functions, and $\|f\|_{p}$, for $f\in\mathcal{L}^{p}(\mu)$, is the $p$-seminorm.

(i) The case $p=1$. If $p=1$, a function $f:X\to\mathbb{R}$ is $p$-integrable if and only if it is integrable: the set of Power-Integrable Functions and the p-Seminorm (ii) is the set $\mathcal{L}^{1}(\mu)$ of integrable functions. For every $f\in\mathcal{L}^{1}(\mu)$,

$$\|f\|_{1}=\int|f|\,d\mu,$$

the integrals of $|f|$ as an element of $\mathcal{M}^{+}$ and as an element of $\mathcal{L}^{1}(\mu)$ being equal.

(ii) Integrability of $|f|^{p}$. A function $f:X\to\mathbb{R}$ belongs to $\mathcal{L}^{p}(\mu)$ if and only if $f$ is measurable and $|f|^{p}\in\mathcal{L}^{1}(\mu)$. In that case the integrals of $|f|^{p}$ as an element of $\mathcal{M}^{+}$ and as an element of $\mathcal{L}^{1}(\mu)$ are equal, and

$$\|f\|_{p}^{p}=\int|f|^{p}\,d\mu .$$

(iii) Homogeneity. Let $f\in\mathcal{L}^{p}(\mu)$ and $c\in\mathbb{R}$. Then $cf\in\mathcal{L}^{p}(\mu)$ and

$$\|cf\|_{p}=|c|\,\|f\|_{p} .$$

(iv) Absolute value. Let $f\in\mathcal{L}^{p}(\mu)$. Then $|f|\in\mathcal{L}^{p}(\mu)$ and $\bigl\||f|\bigr\|_{p}=\|f\|_{p}$.

(v) Domination. Let $f\in\mathcal{L}^{p}(\mu)$, and let $g:X\to\mathbb{R}$ be measurable with $|g(x)|\le|f(x)|$ for almost every $x\in X$. Then $g\in\mathcal{L}^{p}(\mu)$ and $\|g\|_{p}\le\|f\|_{p}$.

(vi) Almost-everywhere equality. Let $f\in\mathcal{L}^{p}(\mu)$, and let $g:X\to\mathbb{R}$ be measurable with $g(x)=f(x)$ for almost every $x\in X$. Then $g\in\mathcal{L}^{p}(\mu)$ and $\|g\|_{p}=\|f\|_{p}$.

(vii) Vanishing seminorm. Let $f\in\mathcal{L}^{p}(\mu)$. Then $\|f\|_{p}=0$ if and only if $f(x)=0$ for almost every $x\in X$.

(viii) Bounded functions. Suppose that $\mu$ is finite. Let $f:X\to\mathbb{R}$ be measurable, and let $B\in\mathbb{R}$ with $|f(x)|\le B$ for every $x\in X$. Then $f\in\mathcal{L}^{p}(\mu)$.