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Let $f: \mathbb{R} \rightarrow [0,+\infty)$ a positive measurable function. Suppose $f$ is improper integrable. Show that $f$ is integrable.

Take a row $x_{n} \rightarrow \infty$ and a row $y_{n}\rightarrow -\infty$ We know that: $\lim_{n\rightarrow \infty}\int_{x_{n}}^{y_{n}} {f} {d\lambda}$ exists. We can write this as: $\lim_{n\rightarrow \infty}\int_{\mathbb{R}} \mathbb{1}_{[x_{n},y_{n}]}f $. We can use now the monotone convergence theorem. Is this OK? Thanks.

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The proof is correct if you assume that both $x_n$ and $y_n$ are monotone. You could actually fix them: $x_n = -n$, $y_n = n$.

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  • $\begingroup$ But we know that $f$ is improper integrable. So we have to go to $+\infty $ and$ -\infty$ at a indepent way. Or am I wrong? $\endgroup$ – bob Jan 7 '16 at 9:36
  • $\begingroup$ We can but not have to. $\endgroup$ – Emanuele Paolini Jan 7 '16 at 9:37
  • $\begingroup$ OK, so we know that it works for every row to $+\infty$ and every row to $-\infty$ (monotone rows) because $f$ is improper integrable. So we can take that row , to apply the monotone convergence theorem. Is this OK? $\endgroup$ – bob Jan 7 '16 at 9:46
  • $\begingroup$ yes, it is correct. $\endgroup$ – Emanuele Paolini Jan 7 '16 at 9:49

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