An improvement of the Kolmogorov-Riesz compactness theorem

The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces

The Kolmogorov-Riesz compactness theorem

We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.Comment: This version is lightly revised in response to referee comments. The paper will appear in Expositiones Mathematica

The generaters of positive semigroups.

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