# Kolmogorov-Riesz theorem

I will show an ${L^1}$ 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 ${\varepsilon>0}$, the set can be covered by finitely many open balls of radius ${\varepsilon}$.

Lemma In a metric space, a set ${E}$ is compact if and only if ${E}$ is complete and totally bounded.

Proof of the lemma can be find here.

Theorem 1 (Kolmogorov Riesz Theorem (finite measure domain)) Let ${\mathcal{F}}$ be a bounded set of ${L^1({\mathbb R}^n)}$ functions, Assume that ${\forall \varepsilon >0}$, there exists a ${\delta >0}$ such that

$\displaystyle \| f(\cdot + y) - f \|_{L^1({\mathbb R}^n)} \le \varepsilon$

whenever ${|y| < \delta}$, for all ${f \in \mathcal{F}}$. Then ${\mathcal{F}_{| \Omega}}$ has a compact closure in ${L^1(\Omega)}$ for any finite measure set ${\Omega \subset {\mathbb R}^n}$.