measure theory - 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.