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

$$(f_n)$$ 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.

• I think the Dominated Convergence Theorem here just works where as dominating function you could take $f_N + 1$ for a large enough $N$ such that this is actually an upper bound. – Stan Tendijck Feb 21 at 0:44
• Do not use any theorem. Just use definitions as in the answer below. – Kavi Rama Murthy Feb 21 at 6:42

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