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

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

(Definition)

Let be a nonempty set and $\mathcal{A}$ be a $\sigma$ -algebra on . Also, let $\mu$ be a non-negative measure defined on $\mathcal{A}$ . Two complex valued functions and are said to be equal almost everywhere on (denoted as a.e. if $\mu \{x \in X : f(x) \ne g(x) \} = 0.$ The relation of being equal almost everywhere on defines an equivalence relation. It is a common practice in the integration theory to denote the equivalence class containing by itself. It is easy to see that if are equivalent and are equivalent, then are equivalent, and 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

$\displaystyle \left\lVert f\right\rVert _{\text{ess}} = \operatorname{inf}\{M > 0 \colon \mu \{x : \vert f(x)\vert > M\} = 0\},$

called the essential supremum of $\vert f\vert$ on

. Now we define,

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

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

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

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.
$L^{\infty}(X,\mu)$ is the dual of $L^1(X,\mu)$ if is $\sigma$ -finite.
$L^{\infty}(X,\mu)$ is closed under pointwise multiplication, and with this multiplication it becomes an algebra. Further, $L^{\infty}(X,\mu)$ is also a -algebra with the involution defined by $f^*(x) = \overline{f(x)}$ . Since this -algebra is also a dual of some Banach space, it is called von Neumann algebra.

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Cross-references: von Neumann algebra, involution, algebra, pointwise, closed under, Banach space, complete, metric, norm, normed linear space, essential supremum, classes, multiplication, addition, equivalent, easy to see, equivalence class, theory, equivalence relation, relation, almost everywhere, functions, complex, measure
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This is version 8 of $L^{\infty}(X, \mu)$ , born on 2003-10-15, modified 2007-08-15.
Object id is 4797, canonical name is LinftyXDmu.
Accessed 2435 times total.

Classification:

AMS MSC:

28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions)

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