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Let $g$, $\{g_n\}$ be integrable and $f$, $\{f_n\}$ be measurable. Suppose $\mid{f_n}\mid\le{g_n}$ and $f_n\to f \ a.e$ and $\lim_{n \to \infty} \int{g_n}=\int g$. Show that $\lim_{n \to \infty} \int | f_n-f| =0$.

I will use dominated convergence theorem. But I can't find the dominating function.

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  • $\begingroup$ You could have used a better title like "Variant of the dominated convergence theorem". $\endgroup$ – aexl May 15 '15 at 18:55
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    $\begingroup$ And I think you some more requirements, something like $g_n \to g$ a.e. $\endgroup$ – aexl May 15 '15 at 19:03
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    $\begingroup$ If you add the requirement $g_n \to g$ ae., then you can find an answer here math.stackexchange.com/q/72174/27978. $\endgroup$ – copper.hat May 15 '15 at 19:13
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This is false (as @Alex S suggested). Working on $\mathbb {R}$ with Lebesgue measure, set $f_n=g_n = \chi_{[n,n+1]}.$ Then $f_n \to 0$ everywhere, and $\int g_n \to \int \chi_{[0,1]}.$ But clearly $f_n$ does not convverge to $0$ in $L^1.$

Let's add the hypothesis that $g_n \to g$ a.e. Then hint: Apply Fatou's Lemma to $g+g_n -|f-f_n|.$

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  • $\begingroup$ Yes! if this condition add, i solve this problem. Thank you! $\endgroup$ – asdwewqe May 15 '15 at 19:24
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Let $f_n = 1_{[n,n+1]}, f = 0$, $g_n = f_n$ and $g=1_{[0,1]}$, then we have $f_n(x) \to f(x)$ everywhere, $|f_n| \le g_n$, $\int g_n = \int g$, but $\int |f_n-f| = 1$ for all $n$.

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  • $\begingroup$ Thank you! i know that this problem needs more requirement. $\endgroup$ – asdwewqe May 15 '15 at 19:24

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