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In the paper we make an effort to answer the question ``What if Arzel\`a andAscoli lived long enough to see Pego theorem?''. Giulio Ascoli and CesareArzel\`a died in 1896 and 1912, respectively, so they could not appreciate thecharacterization of compact families in $L^2(\mathbb{R}^N)$ provided by RobertL. Pego in 1985. Unlike the Italian mathematicians, Pego employed various toolsfrom harmonic analysis in his work (for instance the Fourier transform or theHausdorff-Young inequality). Our article is meant to serve as a bridge betweenArzel\`a-Ascoli theorem and Pego theorem (for $L^1(G)$ rather than $L^2(G)$,$G$ being a locally compact abelian group). In a sense, the former is the``raison d'\^etre'' of the latter, as we shall painstakingly demonstrate.

How Arzel\`a and Ascoli would have proved Pego theorem for $L^1(G)$ (if they lived in the $21^{st}$ century)?