Characterization of the classical extensions of the family of continuous functions as Dedekind hulls

Abstract

In various fields of mathematics, extensions of the family C of all bounded continuous functions on a topological space T arise; many of them have become classical. Such extensions include the families BM and BM 0 of functions measurable in the sense of Borel and Baire, the families B and B 0 of classes of functions with the Baire property and with the Baire property relative to cozero sets, the family R µ of classes of Riemann µ -integrable functions, the family L µ of classes of Lebesgue µ -measurable functions, the family UM of universally measurable functions, etc.