Question on Real Analysis

Asked Apr 19 '12 at 04:40

Active Apr 19 '12 at 04:40

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Possible Duplicate:
convergence in $L^1$ norm
Convergence of Lebesgue integrals

let $f$ and $f_n,n\geq 1$ be integrable functions on $X$. Suppose that $f_n \to f$ a.e. on $X$ as $n\to \infty$, and $\int_X f_n d\mu \to \int_X fd\mu$ as $n\to \infty$. Show that if every $f_n \geq 0$ a.e. on $X$, then $$ \lim_{n\to\infty} \int_X |f_n -f|d\mu = 0 .$$

Please how do I begin? Thanks

edited Apr 13 '17 at 12:21

Community

asked Apr 19 '12 at 04:40

Dan

1

See here. But for just a hint: Apply Fatou's Lemma to the sequence with terms $f_k+f-|f_k-f|$. – David Mitra Apr 19 '12 at 04:45
@DavidMitra. Thanks for the link. But I don't know if $f \gt 0$. I only know that $f_n \gt 0$. – Dan Apr 19 '12 at 04:51
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@Dan If $f_n\ge 0$ a.e. for each $n$ and if $f_n\rightarrow f$ a.e., then $f\ge0$ a.e.. – David Mitra Apr 19 '12 at 04:53
@DavidMitra That was my concern, Thank you. – Dan Apr 19 '12 at 04:56
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I think this is and exact duplicate of this – leo Apr 19 '12 at 05:06
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And all this sort of questions can be answered by this one – leo Apr 19 '12 at 05:08

Question on Real Analysis

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