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Let $(f_{n})_{n}$ a sequence in $\mathcal{L}^1(\mathbb{R})$ and $f_{n}<f_{n+1}$. Also $\int_{\mathbb{R}}f_{k}^-dm < \infty$ for some $k\in \mathbb{N}$. Show that $$\lim_{n\to\infty}\int_{\mathbb{R}}f_{n}dm=\int_{\mathbb{R}}\lim_{n\to\infty}f_{n}dm.$$ I think write this like a growing sequence for use the monotone Lebesgue theorem, some help? Thanks! |
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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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