Remarks on the Glivenko-Cantelli theorem in a separable metric space
The authors prove the Glivenko-Cantelli theorem for separable metric spaces. Furthermore, they obtain the following result: Let \(X\) be a random element in a separable metric space \(T\), \(\mathcal{U}\) an arbitrary class of Borel sets in \(T\), \(\delta >0\), and \(\partial_\delta(U)\) the \(\delta\)-neighbourhood of the border \(\partial U\) of a set \(U\). Then there exists a system of Borel sets \(\mathcal{U'}\) with the following condition: \(\forall\, U\in\mathcal{U}\;\;\;\exists\,U'\in \mathcal{U'}\,,\;\; U\subset U'\,:\;U'\backslash U\subset\partial_\delta(U)\), for which the Glivenko-Cantelli theorem is valid.
scientific article; zbMATH DE number 6212603