Let $(f_{n})_{n}$ a sequence in $\mathcal{L}^1(\mathbb{R})$ and $f_{n} I think write this like a growing sequence for use the monotone Lebesgue theorem, some help? Thanks!
Convergence of Lebesgue integral with negative part finite
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integration
measure-theory
convergence
1 Answers
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If you assume that the limit exists almost-everywhere (otherwise the question doesn't even make sense), you basically spelled out the answer.
Without loss of generality, the negative part of $f_0$ is integrable (we can always forget finitely many elements of the sequence). Put $g_n=f_n+(f_0^-)$. Then $g_n$ are integrable, positive and increasing, so by monotone convergence $$\int f_n+\int f_0^-=\int g_n\to \int\lim g_n=\int(\lim f_n+f_0^-)=\int\lim f_n+\int f_0^-$$ from which you immediately get the result.
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3The pointwise limit of an increasing sequence of functions always exists in the extended reals. – 2012-06-14
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0You can use `\left(` and `\right)` to adjust parenthesis size. – 2012-06-14
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1M. Greinecker: you're right. It still makes sense, because all the functions considered are (mostly) positive. Still, I hesitate to write an integral of a function infinite on a big set. – 2012-06-14
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0did: I know, but doing that wouldn't improve readability much in this case, and would make the expression two-line, which would be bad. Instead, I changed the notation somewhat -- should be prettier now. – 2012-06-14