Remarks on the Glivenko-Cantelli theorem in a separable metric space
type: Item
instance of: scholarly article
zbMATH Open document ID: 1289.60001
author: I. K. Matsak
author: A. M. Plichko
publication date: 2013-09-26T00:00:00Z
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 6212603
zbMATH Keywords: Glivenko-Cantelli theorem
zbMATH Keywords: separable space
MaRDI profile type: Publication
title: Remarks on the Glivenko-Cantelli theorem in a separable metric space
published in: Matematychnyĭ Visnyk Naukovogo Tovarystva Im. Shevchenka
review text: 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.
description: scientific article; zbMATH DE number 6212603