Complete the proof of Corollary 9.9 and show that (9.6) is actually equivalent to (9.5) in Beppo

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 Complete the proof of Corollary 9.9 and show that (9.6) is actually equivalent to (9.5) in Beppo Levi's theorem.

Data from corollary 9.9

18 Let (Un)neNCM(A). Then 1 un is measurable and we have n=1 (9.6)  h=1 = / to the un du = (including the

Data from theorem 9.6

(Beppo Levi) Let (X, A, ) be a measure space. For an increasing sequence of functions (Un)neN CME(A), 0un <

Step 1. Claim: u, w M (A), u

M where we use the definition of the sets B (for the penultimate estimate) and the fact that 1 B, < 1 (in the

Step 4. In the estimate which we established in Step 3, we can go to the supremum over all fe&t () with f

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