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Let $\{g_k\}$ and $g$ be integrable functions, $\{f_k\}$ and $f$ measurable functions, and $|f_k|\le g_k$, $f_k\to f$ almost everywhere If $$\lim_{k\to\infty} \int g_k\ \mathsf d\mu=\int g\ \mathsf d\mu,$$

Prove or disprove that $$\lim_{k\to\infty} \int |f_k-f|\ \mathsf d\mu=0. $$

I tried to approach this problem by using similar method as proof of dominated convergence theorem but it failed.

How can I approac this problem properly?

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It's false:

Consider $[0,1]$ with Lebesgue measure. Let $f_k(x)=g_k(x)=n\chi_{[0,1/n]}$. Let $g=1$ and let $f=0$.

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  • $\begingroup$ Thankyou very much. i have one more question, if $\lim g_k=g$ condition added, does the result of the problem changes? $\endgroup$
    – Chesed
    Nov 27, 2015 at 8:09
  • $\begingroup$ @Moon Is that convergence p.w.a.e.? $\endgroup$
    – user223391
    Nov 27, 2015 at 8:10
  • $\begingroup$ it is pointwise $\endgroup$
    – Chesed
    Nov 27, 2015 at 8:10

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