Convergence and Lebesgue Integration I came across this question in a textbook on introductory Lebesgue Integration. I have been teaching myself this material but was unsure of how to do the following question:
Let $(g_n)$ be a sequence of functions that is uniformly bounded and converges pointwise to $g$ almost everywhere. Show that $(g_n)$ converges in the mean to $g$. 
I am working with the following definition of convergence in the mean:
$$\lim \int_I{|f_n −f|\ d\mu} = 0$$
Any help would be greatly appreciated. Thank you.

It is true when $I$ is bounded, either by dominated convergence or Egoroff's theorem.
However, when $I=\Bbb R_+$, it may be not true, for example if $f_n=\chi_{(n,n+1)}$ ($|f_n|\leqslant 1$ for all $n$, and $f_n\to 0$ almost everywhere but $\int_I|f_n-f|=1$). 
