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On Characterizations of Conditional Expectation
Published online by Cambridge University Press: 20 November 2018
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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).
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- Research Article
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- Copyright © Canadian Mathematical Society 1973
References
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Neveu, J., Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, Calif., 1965.Google Scholar
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