Almost-Everywhere Equality on the p-Integrable Functions

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\ge1$ be a real number. $\mathcal{L}^{p}(\mu)$ is the set of $p$-integrable functions and $\|f\|_{p}$, for $f\in\mathcal{L}^{p}(\mu)$, is the $p$-seminorm. Let $\sim$ be the relation on $\mathcal{L}^{p}(\mu)$ given by

$$f\sim g\iff f(x)=g(x)\ \text{for $\mu$-almost every }x\in X\qquad(f,g\in\mathcal{L}^{p}(\mu)).$$

(i) Equivalence relation. $\sim$ is an equivalence relation on $\mathcal{L}^{p}(\mu)$.

(ii) Compatibility with the operations. Let $f,f',g,g'\in\mathcal{L}^{p}(\mu)$ with $f\sim f'$ and $g\sim g'$, and let $c\in\mathbb{R}$. Then $f+g\sim f'+g'$ and $cf\sim cf'$.

(iii) The seminorm. Let $f,g\in\mathcal{L}^{p}(\mu)$ with $f\sim g$. Then $\|f\|_{p}=\|g\|_{p}$.

(iv) The integral of a product. Suppose that $p=2$, and let $f,f',g,g'\in\mathcal{L}^{2}(\mu)$ with $f\sim f'$ and $g\sim g'$. Then $fg,f'g'\in\mathcal{L}^{1}(\mu)$ and

$$\int fg\,d\mu=\int f'g'\,d\mu .$$