Question for dominated convergence theorem.

Question

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?

Answer 1

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 $f=0$.