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uto2rimxrs50

uto2rimxrs50

Answered question

2022-05-11

Let E R be a set of finite measure. Assume the sequence f n is in L 1 ( E ) and f n f weakly in L 1 ( E ). Show that if f n M almost everywhere for a constant M then f M almost everywhere.
I tried providing it by contradiction, supposing f < m on a subset of E with positive measure. Then of course for every n N , there is a subset A n of A such that f n M on A n , but I can't go any further.

Answer & Explanation

pradassas66b2d

pradassas66b2d

Beginner2022-05-12Added 11 answers

Let F = { x E : f ( x ) < M }. Then F f n F f because χ F L . Hence F f M μ ( F ). Can you finish?

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