Definition ($\mathscr{L}^{p}$ space)

For $1\le p<\infty$ we define $$\mathscr{L}^{p}(X,\mathcal{F},\mu)=\left\{ f:X\to \mathbb{R}\text{ measurable }: \int\limits _{X}|f|^p \, d\mu<\infty \right\}$$ Let the relation $\sim$ on $L^{p}(X,\mathcal{F},\mu)=\frac{\mathscr{L}^{p}(X,\mathcal{F},\mu)}{\sim}$ be defined $\forall f,g\in\mathscr{L}^p(X,\mathcal{F},\mu)$ as follows: $$f\sim g\iff f=g \ \mu\text{-a.e.}$$