L-infinity function - Knowlpedia

A $L^\infty$ function on a measure space $(X,\Sigma,\mu)$ is a measurable function $f:X\to\mathbb R$ or $f:X\to\mathbb C$ such that

\|f\|_\infty := \operatorname*{ess\,sup}_{x\in X} |f(x)| < \infty.

Here $\operatorname*{ess\,sup}$ denotes the essential supremum.

If $f$ and $g$ are equal almost everywhere, then $\lVert f\rVert_\infty=\lVert g\rVert_\infty$ , so membership in $L^\infty$ depends only on the equivalence class modulo a null set. These equivalence classes form the $p=\infty$ case of an $L^p$ space.

Examples

On $([0,1],\mathcal B,\lambda)$ , the function $f(x)=x$ is in $L^\infty$ and satisfies $\lVert f\rVert_\infty=1$ .
If $A$ is a measurable set in $X$ , then the indicator function $\mathbf 1_A$ is in $L^\infty$ and $\lVert\mathbf 1_A\rVert_\infty\le 1$ , with equality to $0$ when $A$ is a null set.