Complete the proof of Theorem 27.11. Data from theorem 27.11 Theorem 27.11 Let (X, A,p) be a

Complete the proof of Theorem 27.11.

Data from theorem 27.11

Transcribed Image Text:

Theorem 27.11 Let (X, A,p) be a o-finite measure space and let HCGCA be sub-o-algebras. The conditional expectation E has the following properties (u, wEL (A), p = [1, ], a, BER):4 (1) Eu is 9 measurable; (ii) u L(A) Eu=L(G) and Eup < ||u||pi (iii) (Eu, w) = (u, Ew) = (Eu, Ew) (for u, wEL (A), UE wEL(A), e.g. if u LP (A) and we L (A) with p+q=1); (iv) u=w Eu=Ew; (v) E (au+Bw)=aEu + BE w;