Proof of Hölder's Inequality

Throughout, $fg$ and $|fg|$ are measurable (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)), $|fg|\in\mathcal{M}^{+}$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv)), integrals of nonnegative functions are taken in $\mathcal{M}^{+}$, and products in $[0,\infty]$ of real numbers are the products in $\mathbb{R}$ (Addition and Multiplication on [0,∞] (ii)). By Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv), $fg\in\mathcal{L}^{1}(\mu)$ as soon as $\int|fg|\,d\mu<\infty$.

Hölder's inequality. Write $A=\|f\|_{p}$ and $B=\|g\|_{q}$.

Case $A=0$ or $B=0$. Say $A=0$; the case $B=0$ is the same with $f$ and $g$ exchanged. By Elementary Properties of the p-Seminorm (vii), $f(x)=0$ for almost every $x$, and at each such $x$ also $|f(x)g(x)|=0$; so $|fg|=0$ almost everywhere (Null Sets and Properties Holding Almost Everywhere (vi)), and $\int|fg|\,d\mu=0$ by The Lebesgue Integral and Null Sets (iv). Hence $fg\in\mathcal{L}^{1}(\mu)$ and $\int|fg|\,d\mu=0\le AB$.

Case $A,B>0$. Let $x\in X$. Young's inequality (Conjugate Exponents and Young's Inequality (ii)) with $a=|f(x)|/A$ and $b=|g(x)|/B$, together with $a^{p}=|f(x)|^{p}/A^{p}$ and $b^{q}=|g(x)|^{q}/B^{q}$ (Properties of Real Powers (v)), gives

$$\frac{1}{AB}\,|f(x)g(x)|\le\frac{1}{pA^{p}}\,|f(x)|^{p}+\frac{1}{qB^{q}}\,|g(x)|^{q} .$$

The three coefficients are positive real numbers. Integrating, by Basic Properties of the Integral of Nonnegative Measurable Maps (iv), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i) and Basic Properties of the Integral of Nonnegative Measurable Maps (iii), and using $\int|f|^{p}\,d\mu=A^{p}$ and $\int|g|^{q}\,d\mu=B^{q}$ (Elementary Properties of the p-Seminorm (ii)), we get

$$\frac{1}{AB}\int|fg|\,d\mu\le\frac{1}{pA^{p}}\int|f|^{p}\,d\mu+\frac{1}{qB^{q}}\int|g|^{q}\,d\mu=\frac{1}{p}+\frac{1}{q}=1 .$$

If $\int|fg|\,d\mu$ were $\infty$, the left side would be $\infty$ (Addition and Multiplication on [0,∞] (ii)); so $\int|fg|\,d\mu$ is a real number (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii)), $fg\in\mathcal{L}^{1}(\mu)$, and multiplying by $AB>0$ gives

$$\int|fg|\,d\mu\le\|f\|_{p}\,\|g\|_{q} .$$

The Cauchy-Schwarz inequality. The exponents $p=q=2$ satisfy $1/2+1/2=1$, so (i) gives $fg\in\mathcal{L}^{1}(\mu)$ and $\int|fg|\,d\mu\le\|f\|_{2}\|g\|_{2}$. By Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (x), $|fg|\in\mathcal{L}^{1}(\mu)$ and $\bigl|\int fg\,d\mu\bigr|\le\int|fg|\,d\mu$, the integral of $|fg|$ as an element of $\mathcal{L}^{1}(\mu)$; it equals the integral in $\mathcal{M}^{+}$ by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi). Hence

$$\Bigl|\int fg\,d\mu\Bigr|\le\|f\|_{2}\,\|g\|_{2} .$$