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Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that $$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$

Proposition: $f$ is integrable over $\mathbb{R}$ and $$\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm.$$

I'm having trouble with proving the integrability. Once that has been shown, I can complete the proof by applying the dominated convergence theorem and using the appropriate indicator function.

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    If $f=\chi_{x>0}-\chi_{x<0}$, we don't have the result.2012-11-01
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    Or even f(x)=x...2012-11-01
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    I figured something could be wrong with the hypothesis. But what if $$\lim_{n \to \infty} \int^n_{-n}|f|dm < \infty?$$2012-11-01
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    In this case, it will work. Have you tried a proof?2012-11-01
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    Let $f_n=f \chi _{[-n,n]}$ and consider $f_n^+$ and $f_n^-$. The functions in the latter two sequences are non-negative, measurable and increase monotonically to $f^+$ and $f^-$, respectively. By the monotone convergence theorem $$\int f^+dm=\lim_{n \to \infty} \int f_n^{+} dm< \infty$$ and likewise for the other function. Is that valid?2012-11-02

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