Suppose $\{f_n\}$ converges to $f$ almost everywhere and $f_n$ are nonnegative measurable functions.

Asked Oct 10 '19 at 19:53

Active Oct 11 '19 at 00:59

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Suppose $\{f_n\}$ converges to $f$ almost everywhere and $f_n$ are nonnegative measurable functions. If \begin{equation*} \lim_{n\rightarrow\infty}\int_Ef_n(x)d\mu=\int_Ef(x)d\mu, \end{equation*} then for any measurable subset $F\subseteq E$, \begin{equation*} \lim_{n\rightarrow\infty}\int_Ff_n(x)d\mu=\int_Ff(x)d\mu. \end{equation*}

How do I prove this? What kind of theorem should I use? Thanks.

edited Oct 11 '19 at 00:59

Math1000

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asked Oct 10 '19 at 19:53

2

Hint. Use Fatou's lemma first to show that $$\limsup_{n\to\infty} \int_{E}|f_n(x)-f(x)| , \mathrm{d}\mu = 0. $$ – Sangchul Lee Oct 10 '19 at 19:59
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Fatou's Lemma on $F$ and $F^c$. – Dzoooks Oct 10 '19 at 20:03
@cmk Thanks for pointing it out. I edited the problem. – Oct 10 '19 at 20:06
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https://math.stackexchange.com/questions/3386401/almost-complete-proof-that-int-a-f-n-to-int-a-f – user284331 Oct 11 '19 at 01:23
@SangchulLee I managed to show $\lim\sup\int_E|f_n(x)-f(x)|d\mu=0$. Then how do I proceed? Thanks. – Oct 11 '19 at 13:04
Can you show the following inequality? $$\left|\int_{\color{blue}F}(f_n-f),\mathrm{d}\mu\right|\leq\int_{\color{red}E}|f_n-f|,\mathrm{d}\mu$$ – Sangchul Lee Oct 11 '19 at 16:17

Suppose $\{f_n\}$ converges to $f$ almost everywhere and $f_n$ are nonnegative measurable functions.

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