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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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    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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With the assumption $\left(\int_{[-n,n]}f\mathrm dm\right)_{n\geqslant 1}$ is convergent, the result does not hold (take and odd non-integrable function).

However, if we assume that $\left(\int_{[-n,n]}|f|\mathrm dm\right)_{n\geqslant 1}$ is convergent, the integrability of $f$ follows from Fatou's lemma.

In the OP's argument, we have to be sure that $\left(\int_{[-n,n]}f^+\mathrm dm\right)_{n\geqslant 0}$ is convergent, but it's not a priori clear.