Proof of Elementary Properties of the p-Seminorm

Throughout, for a measurable $f:X\to\mathbb{R}$ the function $|f|^{p}$ is measurable with nonnegative real values and lies in $\mathcal{M}^{+}$ (Power-Integrable Functions and the p-Seminorm (i) and Real Powers of the Absolute Value of a Measurable Function Are Measurable (i)), and real powers obey Properties of Real Powers. By Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi), a measurable function with nonnegative real values lies in $\mathcal{L}^{1}(\mu)$ if and only if its integral as an element of $\mathcal{M}^{+}$ is finite, and then its two integrals are equal; unless said otherwise, integrals of nonnegative functions below are taken in $\mathcal{M}^{+}$.

Integrability of $|f|^{p}$. By Power-Integrable Functions and the p-Seminorm (ii), $f\in\mathcal{L}^{p}(\mu)$ means that $f$ is measurable and $\int|f|^{p}\,d\mu<\infty$; by the remark above this holds if and only if $f$ is measurable and $|f|^{p}\in\mathcal{L}^{1}(\mu)$, and then the two integrals of $|f|^{p}$ agree. With $I=\int|f|^{p}\,d\mu\ge0$, Power-Integrable Functions and the p-Seminorm (iii) and Properties of Real Powers (vii) give $\|f\|_{p}^{p}=(I^{1/p})^{p}=I$.

The case $p=1$. By Properties of Real Powers (i), $|u|^{1}=|u|$ for every real $u$, so $|f|^{1}=|f|$ for every measurable $f:X\to\mathbb{R}$. By Power-Integrable Functions and the p-Seminorm (ii) and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv), a function $f:X\to\mathbb{R}$ is $1$-integrable if and only if it is measurable with $\int|f|\,d\mu<\infty$, that is, if and only if $f\in\mathcal{L}^{1}(\mu)$ (whose elements are measurable by Integrable Functions and Their Lebesgue Integral (ii)). For $f\in\mathcal{L}^{1}(\mu)$ we have $|f|\in\mathcal{L}^{1}(\mu)$ by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (x), so the two integrals of $|f|$ agree, and $\|f\|_{1}=\bigl(\int|f|\,d\mu\bigr)^{1}=\int|f|\,d\mu$ by Properties of Real Powers (i).

Homogeneity. The function $cf$ is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i). For $x\in X$, Properties of Real Powers (v) gives

$$|cf|^{p}(x)=\bigl(|c|\,|f(x)|\bigr)^{p}=|c|^{p}\,|f(x)|^{p},$$

and $|c|^{p}$ is a real number $\ge0$, whose products with elements of $[0,\infty]$ that are real numbers are the products in $\mathbb{R}$ (Addition and Multiplication on [0,∞] (ii)). By Basic Properties of the Integral of Nonnegative Measurable Maps (iv),

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

so $cf\in\mathcal{L}^{p}(\mu)$, and by Properties of Real Powers (v) and Properties of Real Powers (vii),

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

Absolute value. The function $|f|$ is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), and $\bigl||f(x)|\bigr|=|f(x)|$ for $x\in X$, so $\bigl||f|\bigr|^{p}=|f|^{p}$. Hence $|f|\in\mathcal{L}^{p}(\mu)$ and $\||f|\|_{p}=\|f\|_{p}$.

Domination. For $x\in X$ with $|g(x)|\le|f(x)|$ we have $|g(x)|^{p}\le|f(x)|^{p}$: by Properties of Real Powers (vi) if $|g(x)|<|f(x)|$, with equality if $|g(x)|=|f(x)|$. So $|g|^{p}\le|f|^{p}$ almost everywhere, by Null Sets and Properties Holding Almost Everywhere (vi), and The Lebesgue Integral and Null Sets (ii) gives

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

Hence $\int|g|^{p}\,d\mu<\infty$ (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (iii)), so $g\in\mathcal{L}^{p}(\mu)$, and $\|g\|_{p}\le\|f\|_{p}$ by Properties of Real Powers (vi) with the exponent $1/p>0$ (with equality when the two integrals are equal).

Almost-everywhere equality. At every $x\in X$ with $g(x)=f(x)$ we have $|g(x)|\le|f(x)|$ and $|f(x)|\le|g(x)|$, so both inequalities hold almost everywhere, by Null Sets and Properties Holding Almost Everywhere (vi). Applying (v) to $g$ and $f$ gives $g\in\mathcal{L}^{p}(\mu)$ and $\|g\|_{p}\le\|f\|_{p}$; applying it again with the roles of $f$ and $g$ exchanged gives $\|f\|_{p}\le\|g\|_{p}$.

Vanishing seminorm. By Properties of Real Powers (ii), applied with the exponent $1/p$, $\|f\|_{p}=0$ if and only if $\int|f|^{p}\,d\mu=0$. By The Lebesgue Integral and Null Sets (iv), this holds if and only if $|f(x)|^{p}=0$ for almost every $x$. By Properties of Real Powers (ii) again, $|f(x)|^{p}=0$ exactly when $|f(x)|=0$, that is, when $f(x)=0$; so the two properties hold at the same points of $X$, and $|f|^{p}=0$ almost everywhere if and only if $f=0$ almost everywhere (Null Sets and Properties Holding Almost Everywhere (vi)).

Bounded functions. For every $x\in X$, $|f(x)|\le B\le|B|$, so $0\le|f|^{p}(x)\le|B|^{p}$ by Properties of Real Powers (ii) and Properties of Real Powers (vi) (with equality when $|f(x)|=|B|$). The measurable function $|f|^{p}$ is therefore bounded by the real number $|B|^{p}$ on the finite measure space, so $|f|^{p}\in\mathcal{L}^{1}(\mu)$ by Bounded Measurable Functions on a Finite Measure Space Are Integrable (i), and $f\in\mathcal{L}^{p}(\mu)$ by (ii).