Question
Please solve the exercise 3.20 .
Thank you for your help !

Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-me
3.2. Convergence theorems. We now ready to explain a quite powerful theorem that is called the monotone convergence theorem.
Remark 3.17. In the above theorem, {n} should be convergence, where we may consider as the limit (i.e., we regard as a numb
n+00JX If lim / Indu = 00, then (3.2) is trivial. In the following, we may assume lim / fn du < 00. Fix a € (0,1) arbitrary,
(3.4) Since a € (0,1) and 0 x <f(x), we have f(x) > 18.(r)f(x) > 16.(x)ao(x), and thus we get Je su dep 2 / 15.9n du 2 / 15,0
Notice that this holds for arbitrary a € (0,1), Hence, it remains true for a = 1, and thus we obtain Saludin 2 fs This conclu
Lemma 3.19. Let E PL(X, M). Then there exists {0;}}such that OS) 02() S... ;(r) S... <f(1) and lim n(x) = f(r) are valid for
holds for f,9 € PL(X, M). Then, the required property follows from in- duction. Exercise 3.20. Complete the proofs of Lemma 3

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Answer #1

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Please solve the exercise 3.20 . Thank you for your help ! ⠀ Review. Let M...
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