Proof of Almost-Everywhere Equality on the p-Integrable Functions

For functions $u,v:X\to\mathbb{R}$ write

$$E(u,v)=\{x\in X:u(x)=v(x)\},$$

so that, for $u,v\in\mathcal{L}^{p}(\mu)$, $u\sim v$ means that $E(u,v)$ holds $\mu$-almost everywhere (A Property Holding Almost Everywhere (i)). Two facts about almost-everywhere statements are used throughout: if $Q$ and $R$ hold $\mu$-a.e., so does $Q\cap R$, and if $Q$ holds $\mu$-a.e. and $X\cap Q\subseteq R$, then $R$ holds $\mu$-a.e. (Null Sets and Properties Holding Almost Everywhere (vii), Null Sets and Properties Holding Almost Everywhere (vi)).

Equivalence relation. Let $f,g,h\in\mathcal{L}^{p}(\mu)$. Since $E(f,f)=X$ and $X\setminus X=\emptyset$ with $\emptyset\in\mathcal{F}$ and $\mu(\emptyset)=0$ (Measure, Measure Space, and Probability Measure (i)), the set $X\setminus E(f,f)$ is $\mu$-null (Null Set of a Measure (i)), so $f\sim f$. Since $E(f,g)=E(g,f)$, $f\sim g$ implies $g\sim f$. Finally

$$E(f,g)\cap E(g,h)\subseteq E(f,h),$$

so $f\sim g$ and $g\sim h$ imply $f\sim h$. Thus $\sim$ is reflexive, symmetric and transitive, an equivalence relation (Equivalence Relations on a Set (i)).

Compatibility. The functions $f+g$, $f'+g'$, $cf$ and $cf'$ lie in $\mathcal{L}^{p}(\mu)$ by Minkowski's Inequality (i). Since the operations are pointwise,

$$E(f,f')\cap E(g,g')\subseteq E(f+g,f'+g'),\qquad E(f,f')\subseteq E(cf,cf'),$$

so $f+g\sim f'+g'$ and $cf\sim cf'$.

The seminorm. The function $g$ is measurable and $g(x)=f(x)$ for almost every $x$, so $\|g\|_{p}=\|f\|_{p}$ by Elementary Properties of the p-Seminorm (vi).

The integral of a product. By Hölder's Inequality (ii), $fg$ and $f'g'$ lie in $\mathcal{L}^{1}(\mu)$; in particular $f'g'$ is measurable. Since

$$E(f,f')\cap E(g,g')\subseteq E(fg,f'g'),$$

we have $fg=f'g'$ almost everywhere, and $\int fg\,d\mu=\int f'g'\,d\mu$ by The Lebesgue Integral and Null Sets (iii).