I will show an version of Arzela-Ascoli theorem, which is called Kolmogorov-Riesz theorem, by using a fact in topology as following.

DefinitionIn a metric space, a set is totally bounded, if for any fixed , the set can be covered by finitely many open balls of radius .

LemmaIn 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 thatwhenever , for all . Then has a compact closure in for any finite measure set .