# Question for dominated convergence theorem.

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?

Consider $[0,1]$ with Lebesgue measure. Let $f_k(x)=g_k(x)=n\chi_{[0,1/n]}$. Let $g=1$ and let $f=0$.
• Thankyou very much. i have one more question, if $\lim g_k=g$ condition added, does the result of the problem changes? Nov 27, 2015 at 8:09