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This is a problem I'm struggling with.

Let $f \in L$ and $0 \le f < 1$. Show that $\lim_{n\to\infty}\int_{[a,b]} f^n~dx = 0$.

My professor said I should take $f = f_1-f_2$, and set $f_{1k} = \min(f_1,k), f_{2k} = \min(f_2,k)$ for $k \in \mathbb N$. then $f_k=f_{1k}-f_{2k}$ and show that $f_k^n \to f^n$. I don't really understand how the $\min(f_1,k)$ is useful for anything. Thanks

spaceisdarkgreen
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MathNoob
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  • But what are $f_1$ and $f_2$?I don't think the hint is useful. Think if $f^n$ converges somewhere uniformly, then apply a convergence theorem you know. – Sarvesh Ravichandran Iyer Feb 18 '17 at 00:00
  • f1 and f2 are elements of L+ – MathNoob Feb 18 '17 at 00:01
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    if $L$ means $L^1[a,b]$ then you can use the dominated convergence theorem (or monotone convergence theorem) since $0<f<1$ implies $f^n<f$ and $f^n\to 0.$ I don't get the hint at all. What's the point of separating f into positive and negative parts when you know 0<f<1? Also what's the point of comparing to a positive integer. Are you sure about $0 < f < 1$? – spaceisdarkgreen Feb 18 '17 at 00:02
  • @spaceisdarkgreen Same here, I do not get why the split is required! – Sarvesh Ravichandran Iyer Feb 18 '17 at 00:04

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