Geodesic
On Characterizations of Conditional Expectation
Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 161-163

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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). 
Al-Hussaini, A. N. On Characterizations of Conditional Expectation. Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 161-163. doi: 10.4153/CMB-1973-028-9
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     doi = {10.4153/CMB-1973-028-9},
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