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On Characterizations of Conditional Expectation

Published online by Cambridge University Press:  20 November 2018

A. N. Al-Hussaini
A. N. Al-Hussaini*
Affiliation:
University of Alberta, Edmonton Alberta
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In the following (Ω, α, μ) is a totally σ-finite measure space except where noted. For a sub-σ-algebra β ⊂ α, the conditional expectation E{f|β} off given β is a function measurable relative to β, such that

In [5] R.G.Douglas proved, among other things the following, in the finite case:

Suppose μ(Ω)=l. Then a linear operator T on L1(Ω, α,μ) is a conditional expect ion if and only if

1.1

1.2

1.3

The point of this note is to characterize conditional expectation in the σ-finite case (Theorems 2, 3).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 161 - 163
Copyright
Copyright © Canadian Mathematical Society 1973

References

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