L-infinity function - Imported Knowlpedia Content

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

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

Here $\operatorname*{ess\,sup}$ denotes the essential supremum, and $|f(x)|$ uses the absolute value.

If $f$ and $g$ are equal almost everywhere, then $\|f\|_\infty=\|g\|_\infty$ , so “being in $L^\infty$ ” depends only on the function up to changes on a null set. The collection of such functions (modulo a.e. equality) is 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 $\|f\|_\infty=1$ .
If $A$ is a measurable set in $X$ , then the indicator function $\mathbf{1}_A$ is in $L^\infty$ and $\|\mathbf{1}_A\|_\infty\le 1$ (with $\|\mathbf{1}_A\|_\infty=0$ when $A$ is a null set).