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

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Complete the proof of Theorem 27.11.

Data from theorem 27.11

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(vi) GHE*Eu=E* u; (vii) E (gu)=gEu for all measurable g and u= L(A); (viii) Eg=gE 1 for all 9-measurable g;

Proof Note that the extension of Eu for uL(A) or uLP (A) can be obtained as an a.e. limit of a suitable

Finally, if g is measurable, we see using g g =0 that (g+g)Eu-E (gu)- E (gu)=E(gu).

(viii) Since (X, A,) is o-finite, E1= supEN E14 for some exhausting sequence (An)neNCA with An X and (An)

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