Show that $lim \int f_n = \int f$ in a finite measure space

Question

$(f_n)$ is a sequence of integrable functions that converges uniformly to $f$. The question asks to show that if the space has a finite measure, then $lim \int f_n = \int f$. I've tried using the convergence theorems, but it appears to have more to do with using the definition of uniform convegence plus some smart unequality chain. Maybe.

Answer 1

Hint: Assuming your functions are defined on space $X$, argue that $$ \left|\int f_nd\mu-\int fd\mu\right|\le \int|f_n-f|d\mu\le \int \sup_{x\in X}|f_n(x)-f(x)|d\mu $$