Proof of Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions

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. As in Hölder's Inequality, $\lVert h\rVert_{1}=\int_{X}|h|\,d\mu$ for $h\in\mathcal{L}^{1}$, because $t^{1}=t$ for nonnegative $t$ by Properties of Real Powers (ii).

Claim 1. The map $f+g$ is measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps.

Step 1: $f+g$ lies in $\mathcal{L}^{p}$. For $x\in X$, claim 5 of Properties of the Absolute Value in an Ordered Field gives $|f(x)+g(x)|\le|f(x)|+|g(x)|$, and the sum of two nonnegative reals is at most twice their maximum. Writing $a=|f(x)|$ and $b=|g(x)|$ and using Properties of Real Powers (v) and Properties of Real Powers (iii),

$$\bigl(|f(x)+g(x)|\bigr)^{p}\le\bigl(2\max\{a,b\}\bigr)^{p}=2^{p}\bigl(\max\{a,b\}\bigr)^{p}=2^{p}\max\{a^{p},b^{p}\}\le 2^{p}\bigl(a^{p}+b^{p}\bigr),$$

where the middle equality holds because $t\mapsto t^{p}$ is nondecreasing on $\mathbb{R}_{+}$, so that the larger of $a,b$ has the larger power, and the last step because $a^{p}$ and $b^{p}$ are nonnegative. Integrating with the monotonicity, additivity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$\int_{X}|f+g|^{p}\,d\mu\le 2^{p}\Bigl(\int_{X}|f|^{p}\,d\mu+\int_{X}|g|^{p}\,d\mu\Bigr)<\infty ,$$

so $f+g\in\mathcal{L}^{p}$.

Step 2: the inequality for $p=1$. Here $|f+g|\le|f|+|g|$ pointwise, so $\lVert f+g\rVert_{1}=\int_{X}|f+g|\,d\mu\le\int_{X}|f|\,d\mu+\int_{X}|g|\,d\mu=\lVert f\rVert_{1}+\lVert g\rVert_{1}$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion.

Step 3: the inequality for $1<p$. Let $q$ be the exponent conjugate to $p$, so that $1<q$ and $(p-1)q=p$, and note that $p-1$ is positive. Let $u$ be the map $x\mapsto\bigl(|f(x)+g(x)|\bigr)^{p-1}$. By Elementary Properties of the p-Seminorm (vi), applied with the exponents $p-1$ and $q$, whose product is $p$ and is at least $1$, the map $u$ is measurable, lies in $\mathcal{L}^{q}$ because $f+g\in\mathcal{L}^{p}$ by Step 1, and satisfies

$$\lVert u\rVert_{q}=\bigl(\lVert f+g\rVert_{p}\bigr)^{p-1}.$$

By Properties of Real Powers (iv) and Properties of Real Powers (ii), $t^{p}=t^{1}t^{p-1}=t\,t^{p-1}$ for nonnegative $t$, so for every $x$

$$\bigl(|f(x)+g(x)|\bigr)^{p}=|f(x)+g(x)|\,u(x)\le\bigl(|f(x)|+|g(x)|\bigr)u(x)=|f(x)|u(x)+|g(x)|u(x),$$

using that $u(x)$ is nonnegative. Since $u$ is nonnegative, $|f|u=|fu|$ and $|g|u=|gu|$ pointwise, where $fu$ and $gu$ are the pointwise products. Integrating and applying Hölder's Inequality (i) twice, to the pairs $f,u$ and $g,u$ with the conjugate exponents $p,q$,

$$\int_{X}|f+g|^{p}\,d\mu\le\int_{X}|fu|\,d\mu+\int_{X}|gu|\,d\mu\le\bigl(\lVert f\rVert_{p}+\lVert g\rVert_{p}\bigr)\lVert u\rVert_{q}.$$

By Elementary Properties of the p-Seminorm (i) the left-hand side is $(\lVert f+g\rVert_{p})^{p}$, so

$$\bigl(\lVert f+g\rVert_{p}\bigr)^{p}\le\bigl(\lVert f\rVert_{p}+\lVert g\rVert_{p}\bigr)\bigl(\lVert f+g\rVert_{p}\bigr)^{p-1}.$$

If $\lVert f+g\rVert_{p}=0$, the asserted inequality holds because the right-hand side of it is nonnegative. Otherwise $\lVert f+g\rVert_{p}$ is positive, hence so is $(\lVert f+g\rVert_{p})^{p-1}$ by Properties of Real Powers (i); dividing the last display by that positive number and using $t^{p}=t\,t^{p-1}$ once more gives $\lVert f+g\rVert_{p}\le\lVert f\rVert_{p}+\lVert g\rVert_{p}$.

Claim 2. The map $0_{X}$ taking the value $0$ everywhere is measurable by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and $|0_{X}|^{p}=0_{X}$ by Properties of Real Powers (i), whose integral is $0$ by The Lebesgue Integral and Null Sets (ii) applied with the null set $\varnothing$; so $0_{X}\in\mathcal{L}^{p}$. By claim 1 the set $\mathcal{L}^{p}$ is closed under the pointwise sum, and by Elementary Properties of the p-Seminorm (ii) under the pointwise scalar multiple. Hence The Real Vector Space of Real-Valued Functions on a Set (ii) applies and shows that $\mathcal{L}^{p}$ is a linear subspace of the space of all real-valued maps on $X$ and is itself a vector space over $\mathbb{R}$ with zero vector $0_{X}$.

The three displayed properties of the $p$-seminorm are, in order: the fact that $\lVert f\rVert_{p}$ is a nonnegative real number, recorded in Power-Integrable Functions and the p-Seminorm (iii); Elementary Properties of the p-Seminorm (ii); and claim 1.

Claim 3. We induct on $n$. For $n=1$ the finite sum is $f_{1}$ by claim 1 of Properties of Finite Sums of Vectors, and the assertion is an equality. Assume the assertion for $n$, and let $f_{1},\dots,f_{n+1}\in\mathcal{L}^{p}$. By the recursion in claim 1 of Properties of Finite Sums of Vectors,

$$\sum_{k=1}^{n+1}f_{k}=\Bigl(\sum_{k=1}^{n}f_{k}\Bigr)+f_{n+1},$$

which lies in $\mathcal{L}^{p}$ by the inductive hypothesis and claim 1. Applying claim 1 and then the inductive hypothesis,

$$\Bigl\lVert\sum_{k=1}^{n+1}f_{k}\Bigr\rVert_{p}\le\Bigl\lVert\sum_{k=1}^{n}f_{k}\Bigr\rVert_{p}+\lVert f_{n+1}\rVert_{p}\le\sum_{k=1}^{n}\lVert f_{k}\rVert_{p}+\lVert f_{n+1}\rVert_{p}=\sum_{k=1}^{n+1}\lVert f_{k}\rVert_{p},$$

the last equality by the recursion in claim 1 of Properties of Finite Sums. This completes the induction.