# $L^{\infty}(X,\mu)$

Let $X$ be a nonempty set and $\mathcal{A}$ be a $\sigma$-algebra on $X$. Also, let $\mu$ be a non-negative measure defined on $\mathcal{A}$. Two complex valued functions $f$ and $g$ are said to be equal almost everywhere on $X$ (denoted as $f=g$ a.e. if $\mu\{x\in X:f(x)\neq g(x)\}=0.$ The relation of being equal almost everywhere on $X$ defines an equivalence relation. It is a common practice in the integration theory to denote the equivalence class containing $f$ by $f$ itself. It is easy to see that if $f_{1},f_{2}$ are equivalent and $g_{1},g_{2}$ are equivalent, then $f_{1}+g_{1},f_{2}+g_{2}$ are equivalent, and $f_{1}g_{1},f_{2}g_{2}$ are equivalent. This naturally defines addition and multiplication among the equivalent classes of such functions. For a measureable $f\colon X\to\mathbb{C}$, we define

 $\left\lVert f\right\rVert_{\text{ess}}=\operatorname{inf}\{M>0\colon\mu\{x:|f(% x)|>M\}=0\},$

called the essential supremum of $|f|$ on $X$. Now we define,

 $L^{\infty}(X,\mu)=\{f:X\to\mathbb{C}:\left\lVert f\right\rVert_{\text{ess}}<% \infty\}.$

Here the elements of $L^{\infty}(X,\mu)$ are equivalence classes.

## Properties of $L^{\infty}(X,\mu)$

1. 1.

The space $L^{\infty}(X,\mu)$ is a normed linear space with the norm $\left\lVert\cdot\right\rVert_{\text{ess}}$. Also, the metric defined by the norm is complete, making $L^{\infty}(X,\mu)$, a Banach space.

2. 2.

$L^{\infty}(X,\mu)$ is the dual of $L^{1}(X,\mu)$ if $X$ is $\sigma$-finite.

3. 3.

$L^{\infty}(X,\mu)$ is closed under pointwise multiplication, and with this multiplication it becomes an algebra. Further, $L^{\infty}(X,\mu)$ is also a $C^{*}$-algebra (http://planetmath.org/CAlgebra) with the involution defined by $f^{*}(x)=\overline{f(x)}$. Since this $C^{*}$-algebra is also a dual of some Banach space, it is called von Neumann algebra.

Title $L^{\infty}(X,\mu)$ LinftyXmu 2013-03-22 13:59:46 2013-03-22 13:59:46 ack (3732) ack (3732) 11 ack (3732) Definition msc 28A25