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Fubini and martingale theorems in C* -algebras

Published online by Cambridge University Press:  24 October 2008

C. J. K. Batty
C. J. K. Batty
Affiliation:
Mathematical Institute, Oxford
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Extract

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ where

Thus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, by

and f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. by

wehere εω is the unit point of mass at ω.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 57 - 69
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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