[分析] 實分析兩題

[okhunter ()]

1、Let (X,B,μ) be a finite measure space, and let

f:X→R be a measurable function. If f^{n} is integrable

for each n, then show that ∫f^{n}dμ=c (a constant) holds

for all n=1,2,3,... iff f=χ_{A} for some measurable subset

A of X.

(<=) 這個方向是顯然的。

(=>) 想了很久完全沒頭緒，因此想請教該如何下手。

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推 yusd24      :你可以拆成 {f>1}, {f=1}, {f<1}
→ yusd24      :你可以去證明, 極限存在的條件是 c=μ(f=1)
→ okhunter    :謝了，我再試試~