$\Theta \rho ϵ\eta \Pi \alpha \tau \pi$

Suppose that $(X,𝒮,\mu )$ is a measure space, $p\in [1,\infty )$, and $f:X\to 𝔽$ is measurable. Define $${\Vert f\Vert }_p:={\left({\int }_X{\left|f\right|}^{p}d\mu \right)}^{1/p}.$$

Suppose that $p\in [1,\infty )$. The $L^p$-space of $\mu$ is the vector space of equivalence classes of measurable functions $f:X\to 𝔽$ such that ${\Vert f\Vert }_p<\infty$, where functions equal almost everywhere are identified.

Suppose that $f:X\to 𝔽$ is measurable. Define $${\Vert f\Vert }_{\infty }:=\inf \{c\in [0,\infty ]:\left|f\left(x\right)\right|\le c\text{ for almost every }x\in X\}.$$

The $L^{\infty }$-space of $\mu$ is the vector space of equivalence classes of measurable functions $f:X\to 𝔽$ such that ${\Vert f\Vert }_{\infty }<\infty$, where functions equal almost everywhere are identified.