Proof of Elementary Properties of the p-Seminorm

Each result cited is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement above. Throughout, powers with nonnegative base are those of Real Powers (i), and we refer to the claims of Properties of Real Powers for their properties; for $f\in\mathcal{L}^{p}$ the number $\int_{X}|f|^{p}\,d\mu$ is a nonnegative real number by Power-Integrable Functions and the p-Seminorm (ii).

Claim 1. Put $I=\int_{X}|f|^{p}\,d\mu$, a nonnegative real number. Then, by Properties of Real Powers (iv) and Properties of Real Powers (ii),

$$\bigl(\lVert f\rVert_{p}\bigr)^{p}=\bigl(I^{1/p}\bigr)^{p}=I^{(1/p)\cdot p}=I^{1}=I .$$

Claim 2. The map $cf$ is measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. For every $x\in X$ we have $|(cf)(x)|=|c|\,|f(x)|$ by claim 4 of Properties of the Absolute Value in an Ordered Field, so by Properties of Real Powers (iii),

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

that is $|cf|^{p}=|c|^{p}\,|f|^{p}$ pointwise. The number $|c|^{p}$ is a nonnegative real number by Properties of Real Powers (i), so the homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives $\int_{X}|cf|^{p}\,d\mu=|c|^{p}I$, which is finite; hence $cf\in\mathcal{L}^{p}$. Using Properties of Real Powers (iii) and then Properties of Real Powers (iv) and Properties of Real Powers (ii),

$$\lVert cf\rVert_{p}=\bigl(|c|^{p}I\bigr)^{1/p}=\bigl(|c|^{p}\bigr)^{1/p}I^{1/p}=|c|^{p\cdot(1/p)}\,I^{1/p}=|c|\,\lVert f\rVert_{p}.$$

Claim 3. Let $N$ be the null set of points at which $|f(x)|\le|g(x)|$ fails. For $x\notin N$, Properties of Real Powers (v) gives $(|f(x)|)^{p}\le(|g(x)|)^{p}$; hence $|f|^{p}\le|g|^{p}$ almost everywhere. Both are measurable by Power-Integrable Functions and the p-Seminorm (i), so The Lebesgue Integral and Null Sets (iii) yields

$$\int_{X}|f|^{p}\,d\mu\le\int_{X}|g|^{p}\,d\mu<\infty ,$$

so $f\in\mathcal{L}^{p}$. Applying Properties of Real Powers (v) with the positive exponent $1/p$ to these two nonnegative real numbers gives $\lVert f\rVert_{p}\le\lVert g\rVert_{p}$.

Claim 4. The set where $|f|$ and $|g|$ differ is contained in the set where $f$ and $g$ differ, hence is null by The Lebesgue Integral and Null Sets (i). So $|f|\le|g|$ almost everywhere and $|g|\le|f|$ almost everywhere. The first, with claim 3, gives $g\in\mathcal{L}^{p}$ once we know $f\in\mathcal{L}^{p}$; more precisely claim 3 applied with the roles of $f$ and $g$ as stated there gives $g\in\mathcal{L}^{p}$ and $\lVert g\rVert_{p}\le\lVert f\rVert_{p}$, and applied in the opposite direction gives $\lVert f\rVert_{p}\le\lVert g\rVert_{p}$. Hence $\lVert g\rVert_{p}=\lVert f\rVert_{p}$.

Claim 5. Write $I=\int_{X}|f|^{p}\,d\mu$. By Properties of Real Powers (i) the power $I^{1/p}$ vanishes exactly when $I$ vanishes, so $\lVert f\rVert_{p}=0$ if and only if $I=0$. By The Lebesgue Integral and Null Sets (iv), $I=0$ if and only if $(|f(x)|)^{p}=0$ for almost every $x$. By Properties of Real Powers (i) again, $(|f(x)|)^{p}=0$ holds exactly when $|f(x)|=0$, that is exactly when $f(x)=0$. Hence $\lVert f\rVert_{p}=0$ if and only if $f=0$ almost everywhere.

Claim 6. Measurability of $|h|^{r}$ is proved exactly as in Power-Integrable Functions and the p-Seminorm (i): the map $|h|$ is measurable with nonnegative values by claim 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; for a real $c<0$ the set $\{x:c<(|h(x)|)^{r}\}$ is $X$, and for $0\le c$ it equals $\{x:c^{1/r}<|h(x)|\}$ by Properties of Real Powers (vi), a member of $\mathcal{F}$; the criterion in Measurable Functions and the Lebesgue Integral: Standing Notation (iii) applies.

The map $|h|^{r}$ is nonnegative, so its absolute value is itself, and by Properties of Real Powers (iv), for every $x\in X$,

$$\Bigl(\bigl|\,|h|^{r}(x)\bigr|\Bigr)^{s}=\Bigl(\bigl(|h(x)|\bigr)^{r}\Bigr)^{s}=\bigl(|h(x)|\bigr)^{rs}.$$

Hence the two functions $\bigl|\,|h|^{r}\bigr|^{s}$ and $|h|^{rs}$ are equal, so their integrals over $X$ coincide as members of $[0,\infty]$, and one is finite exactly when the other is. Since $|h|^{r}$ and $h$ are both measurable, this says $|h|^{r}\in\mathcal{L}^{s}$ if and only if $h\in\mathcal{L}^{rs}$.

Suppose this holds and write $I=\int_{X}|h|^{rs}\,d\mu$, a nonnegative real number. Then $\bigl\lVert\,|h|^{r}\bigr\rVert_{s}=I^{1/s}$ by Power-Integrable Functions and the p-Seminorm (iii), while by Properties of Real Powers (iv)

$$\bigl(\lVert h\rVert_{rs}\bigr)^{r}=\bigl(I^{1/(rs)}\bigr)^{r}=I^{r/(rs)}=I^{1/s},$$

the exponents being equal because $r\cdot\frac{1}{rs}=\frac{1}{s}$. The two sides therefore agree.