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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

1. Chacon, R.V., Identification of the limit of operator averages, J. Math, and Mech. (1962), 961–968.Google Scholar
2. Chacon, R.V. and Ornstein, D., A general ergodic theorem, Illinois, J. Math. (1960), 153–160.Google Scholar
3. Chacon, R.V. and Krengel, U., Linear modulus of linear operators, Proc. Amer. Math. Soc. (1964), 553–559.Google Scholar
4. Foguel, S.R., The ergodic theory of Markov processses, Van Nostrand, Princeton, N.J., 1969.Google Scholar
5. Douglas, R.G., Contractive projections on an L1space, Pacific J. Math. (1965), 443–462.Google Scholar
6. Neveu, J., Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, Calif., 1965.Google Scholar