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2. Here we present a few properties that facilitate its immediate use. Proposition 1. Let f : Ω → R be a random variable on a probability space (Ω, Σ, P ) and P ⊂ Σ be a partition of Ω (of positive probability for each member). 2. Proof. This is also an easy consequence of the definitions. 4), (EP (f ))(ω) = EAn (f )(ω), ω ∈ An , n ≥ 1, since Ω = ∪n An and each ω in Ω belongs to exactly one An . Therefore the existence of EP (f ) implies that of EAn (f ), n ≥ 1, and hence of EA (f ) or EA (f ). 3)).
The next result contains the positive statements. Proposition 2. Let {Xn , n ≥ 1} be a sequence of random variables on (Ω, Σ, P ) and B ⊂ Σ be a σ-algebra. , and in L1 (P )-mean. n Proof. (i) Replacing Xn by X − X0 ≥ 0, we may assume that the sequence is nonnegative and increasing. , since E B (X0 ) exists. , also the preceding remark). e. e. Since the extreme integrands on either side of the equality are random variables for B, and A in B is arbitrary, the integrands can be identified. It is precisely the desired assertion.
4 Conditioning with densities Proof. If A ⊂ Rn is an interval and ϕ = χA , then ϕ(X) = χX −1 (A) where X : Ω → Rn is the given random vector. Thus if A is expressible n as × [ai , bi ) we then have on writing X = (X1 , . . , Xn ): i=1 E(ϕ(X)) = P ([X ∈ A]) = P (a1 ≤ X1 < b1 , . . ,Xn (x1 , . . , xn ) an Rn χA dFX . Thus (10) holds in this case. By linearity of E(·) and of the integral n on the last line, (10) also holds if ϕ = i=1 ai χAi , a simple function. Then by the monotone convergence theorem (10) is true for all Borel functions ϕ ≥ 0, since every such ϕ is the pointwise limit of a monotone sequence of simple functions.