I will show an version of Arzela-Ascoli theorem, which is called Kolmogorov-Riesz theorem, by using a fact in topology as following.
Definition In a metric space, a set is totally bounded, if for any fixed , the set can be covered by finitely many open balls of radius .
Lemma In a metric space, a set is compact if and only if is complete and totally bounded.
Proof of the lemma can be find here.
Theorem 1 (Kolmogorov Riesz Theorem (finite measure domain)) Let be a bounded set of functions, Assume that , there exists a such that
whenever , for all . Then has a compact closure in for any finite measure set .