Proof of Minkowski's Inequality

Throughout, integrals of nonnegative functions are taken in $\mathcal{M}^{+}$, sums and products in $[0,\infty]$ of real numbers are those of $\mathbb{R}$ (Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii)), and real powers obey Properties of Real Powers.

Linear combinations. By Elementary Properties of the p-Seminorm (iii), $af$ and $bg$ lie in $\mathcal{L}^{p}(\mu)$, so it suffices to show that $f+g\in\mathcal{L}^{p}(\mu)$ for all $f,g\in\mathcal{L}^{p}(\mu)$. The function $f+g$ is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), so $|f+g|^{p}\in\mathcal{M}^{+}$ by Real Powers of the Absolute Value of a Measurable Function Are Measurable (i). Let $x\in X$, and suppose $|g(x)|\le|f(x)|$. Then $|f(x)+g(x)|\le|f(x)|+|g(x)|\le2|f(x)|$, so Properties of Real Powers (vi) and Properties of Real Powers (v) give

$$|f+g|^{p}(x)\le\bigl(2|f(x)|\bigr)^{p}=2^{p}|f(x)|^{p}\le2^{p}\bigl(|f(x)|^{p}+|g(x)|^{p}\bigr);$$

if $|f(x)|\le|g(x)|$, the same bound follows with $f$ and $g$ exchanged. By Basic Properties of the Integral of Nonnegative Measurable Maps (iii), Basic Properties of the Integral of Nonnegative Measurable Maps (iv) and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i),

$$\int|f+g|^{p}\,d\mu\le2^{p}\Bigl(\int|f|^{p}\,d\mu+\int|g|^{p}\,d\mu\Bigr)<\infty ,$$

so $\int|f+g|^{p}\,d\mu<\infty$ (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (iii)) and $f+g\in\mathcal{L}^{p}(\mu)$.

Minkowski's inequality for $p=1$. By Elementary Properties of the p-Seminorm (i), $\|u\|_{1}=\int|u|\,d\mu$ for every $u\in\mathcal{L}^{1}(\mu)$. Since $|f+g|(x)\le|f(x)|+|g(x)|$ for every $x\in X$, Basic Properties of the Integral of Nonnegative Measurable Maps (iii) and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i) give

$$\|f+g\|_{1}=\int|f+g|\,d\mu\le\int|f|\,d\mu+\int|g|\,d\mu=\|f\|_{1}+\|g\|_{1} .$$

Minkowski's inequality for $p>1$. Put $h=f+g$, which lies in $\mathcal{L}^{p}(\mu)$ by (i), and $N=\|h\|_{p}$. If $N=0$, the inequality holds because $\|f\|_{p},\|g\|_{p}\ge0$ (Properties of Real Powers (ii)). Let $N>0$.

The conjugate exponent. Let $q=p/(p-1)$. By Conjugate Exponents and Young's Inequality (i), $q>1$ and $1/p+1/q=1$; moreover $(p-1)q=p$ and $p/q=p-1$.

The function $w=|h|^{p-1}$. Let $w:X\to\mathbb{R}$, $w(x)=|h(x)|^{p-1}$. Since $p-1>0$, $w$ is measurable with values $\ge0$ by Real Powers of the Absolute Value of a Measurable Function Are Measurable (i), applied with $a=p-1$. By Properties of Real Powers (iv), $|w(x)|^{q}=(|h(x)|^{p-1})^{q}=|h(x)|^{p}$ for every $x\in X$, so $|w|^{q}=|h|^{p}$. Hence $w\in\mathcal{L}^{q}(\mu)$ by Power-Integrable Functions and the p-Seminorm (ii), and by Elementary Properties of the p-Seminorm (ii) and Properties of Real Powers (iv),

$$\|w\|_{q}=\Bigl(\int|h|^{p}\,d\mu\Bigr)^{1/q}=\bigl(N^{p}\bigr)^{1/q}=N^{p/q}=N^{p-1}.$$

The estimate. For $x\in X$, Properties of Real Powers (iv) and Properties of Real Powers (i) give

$$|h(x)|^{p}=|h(x)|^{1}\,|h(x)|^{p-1}=|f(x)+g(x)|\,w(x)\le|f(x)|\,w(x)+|g(x)|\,w(x).$$

Integrating by Basic Properties of the Integral of Nonnegative Measurable Maps (iii) and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i), and applying Hölder's inequality (Hölder's Inequality (i)) to $f\in\mathcal{L}^{p}(\mu)$, $w\in\mathcal{L}^{q}(\mu)$ and to $g\in\mathcal{L}^{p}(\mu)$, $w\in\mathcal{L}^{q}(\mu)$, we obtain, with Elementary Properties of the p-Seminorm (ii),

$$N^{p}=\int|h|^{p}\,d\mu\le\int|fw|\,d\mu+\int|gw|\,d\mu\le\bigl(\|f\|_{p}+\|g\|_{p}\bigr)N^{p-1}.$$

Dividing by $N^{p-1}>0$ (Properties of Real Powers (ii)) and using $N^{p}/N^{p-1}=N^{p-(p-1)}=N^{1}=N$ (Properties of Real Powers (iv) and Properties of Real Powers (i)) gives

$$\|f+g\|_{p}=N\le\|f\|_{p}+\|g\|_{p} .$$