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Monotone convergence theorem doesn't require the sequence of functions $f_n$'s to be $L^1$. When $f_n\in L^1$, will its pointwise limit function $f$ also be in $L^1$? Thanks!

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up vote 5 down vote accepted

Not necessarily.

Define $f_n$ as follows.

On $[-n,n]$, set $f_n(x):=1$ and $f_n(x):=0$ elsewhere.

Each $f_n$ is in $L^1$.

But the pointwise limit, which is equal to the constant function $f(x)=1$ on $\mathbb{R}$, is not in $L^1$.

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Not quite, since the monotone convergence theorem includes the case where $\lim_{n \to \infty} \int f_n \ d\mu = \infty$. But if the limit is finite, $f \in L^1$.

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Thank you! Why is $\lim_{n→∞}∫f_n dμ=∞$, if $f_n \in L^1$? – Ethan Jan 27 '13 at 19:04
Why not? @julien provided one example, it's easy to find many others. – Robert Israel Jan 27 '13 at 19:16

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