Let X be a nonempty set and 𝒜 be a σ-algebra on X. Also, let μ be a non-negative measureMathworldPlanetmath defined on 𝒜. Two complex valued functions f and g are said to be equal almost everywhere on X (denoted as f=g a.e. if μ{xX:f(x)g(x)}=0. The relationMathworldPlanetmath of being equal almost everywhere on X defines an equivalence relationMathworldPlanetmath. It is a common practice in the integration theory to denote the equivalence classMathworldPlanetmath containing f by f itself. It is easy to see that if f1,f2 are equivalentMathworldPlanetmathPlanetmathPlanetmath and g1,g2 are equivalent, then f1+g1,f2+g2 are equivalent, and f1g1,f2g2 are equivalent. This naturally defines addition and multiplication among the equivalent classes of such functions. For a measureable f:X, we define


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


Here the elements of L(X,μ) are equivalence classes.

Properties of L(X,μ)

  1. 1.

    The space L(X,μ) is a normed linear space with the norm ess. Also, the metric defined by the norm is completePlanetmathPlanetmathPlanetmathPlanetmath, making L(X,μ), a Banach spaceMathworldPlanetmath.

  2. 2.

    L(X,μ) is the dual of L1(X,μ) if X is σ-finite.

  3. 3.

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

Title L(X,μ)
Canonical name LinftyXmu
Date of creation 2013-03-22 13:59:46
Last modified on 2013-03-22 13:59:46
Owner ack (3732)
Last modified by ack (3732)
Numerical id 11
Author ack (3732)
Entry type Definition
Classification msc 28A25