L-infinity function

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

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 ).