Definability in function spaces
type: Item
instance of: scholarly article
title: Definability in function spaces
zbMATH Open document ID: 1015.28003
author: Alessandro Andretta
author: Alberto Marcone
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: Let \(X\), \(Y\) denote Polish spaces. Let us recall that a~set \(A\subseteq X\) is wedge reducible to \(B\subseteq Y\) if there is a~continuous function \(f:X\to Y\) such that \(A=f^{-1}(B)\). Wedge reducibility is a~preordering and the classes of equivalences induced by this preordering are called wedge degrees. The authors study the wedge degree of a~function space viewed as a~subspace of another function space. They prove that under some appropriate hypothesis the space \(L^q(X,\mu)\) and the space of all simple functions as subsets of \(L^p(X,\mu)\) are \(\Sigma^0_2\)-complete. If \(X\)~is locally compact then for a~reasonable measure~\(\mu\), \(C(X)\cap L^p(X,\mu)\) is \(\Pi^0_3\)-complete. If \(X\)~is not \(\sigma\)-compact then this space is \(\Pi^1_1\)-complete and hence not Borel. The authors consider also spaces of Riemann integrable functions and of differentiable functions. For a~fixed Polish space~\(X\), they endow \(C(X)\) with the compact-open topology which makes it Polish and study various subspaces of~\(C(X)\).
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 1848835
zbMATH Keywords: Borel hierarchy
zbMATH Keywords: projective hierarchy
zbMATH Keywords: wedge reducibility
zbMATH Keywords: wedge degrees
zbMATH Keywords: function spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1848835