The space $L^p(\mu )$ as the quotient by null functions

Let $(X,\mathcal{A},\mu )$ be a measure space.

• For $0<p<\mathrm{\infty }$, define an equivalence relation on ${\mathcal{L}}^p(\mu )$ by $f\sim g⟺f-g\in \mathcal{N}(\mu ),$ and write $L^p(\mu )$ for the set of classes $[f]$.
• For $p=\mathrm{\infty }$, define the same relation on $L^{\mathrm{\infty }}(\mu )$ and again write $L^{\mathrm{\infty }}(\mu )$ for the set of classes.

Thus an element of $L^p(\mu )$ is an almost-everywhere equivalence class of measurable representatives.

When $1\le p\le \mathrm{\infty }$, the displayed set quotient agrees with the usual quotient-vector-space construction of The quotient vector space $V\mathrm{/}W$ and its canonical projection.

For $0<p<1$, the same class notation is used; under its stated choice hypothesis, the later item The $L^p$ distance for $0<p<1$ is a complete translation-invariant metric supplies the metric structure rather than a normed-space structure.

The quotient vector space $V\mathrm{/}W$ and its canonical projection

The $L^p$ distance for $0<p<1$ is a complete translation-invariant metric