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!
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!
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.