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Please help me with this problem!

Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$.

  1. If the measure of $W$ is finite, the integral of $f_n$ on $\Omega$ converges to the integral of $f$ on $W$. (Should I use monotone convergence THM or dominance convergence THM or neither?)

  2. Show by an example that if the finite-measure hypothesis is dropped then the conclusion may fail.

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1 Answer 1

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  1. No need to use "big" theorems: $$\int_\Omega|f_n-f|d\mu\leqslant \mu(\Omega)\sup_{x\in \Omega}|f_n(x)-f(x)|.$$

  2. Consider the set of positive natural numbers and $f_n=n^{-1}(\underbrace{1,\dots,1}_{n\mbox{ times}},0,\dots)$, $f\equiv 0$.

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Thanks so much. I think I get part 1. –  user65214 Apr 13 '13 at 16:54
    
You are welcome. –  Davide Giraudo Apr 13 '13 at 16:54
    
Could you please explain a bit more for part 2? I see the integral of f is zero, but what is the integral of fn? What kind of function is fn? Thank you. –  user65214 Apr 13 '13 at 16:56
    
$f_n$ is a sequence in this case, whose $n$ first term are $n^{-1}$, and all the others $0$. There is uniform convergence to the null sequence, but not in $L^1$. –  Davide Giraudo Apr 13 '13 at 16:59
    
I see... terrific example, thanks again :) –  user65214 Apr 13 '13 at 17:02
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