Inverses of Borel functions

摘要

A classical result from Lusin and Souslin tells us that should a Borel function f : X - Y be a bijection, then its inverse f(-1) : Y - X must also be a Borel function. Let bB = {f : [0, 1] - Q - [0, 1] - Q : f is Borel measurable}. Here, we consider the sets bB' = {f is an element of bB : f is one-to-one}, and bH = {f is an element of bB' : f : [0, 1] - Q - [0,1] - Q is a bijection}, each endowed with the metric d(f,g) = sup(x is an element of[0,1]-Q) vertical bar f (x) - g(x) vertical bar, and the latter also with the metric d* (f, g) = d(f, g) + d(f(-1), g(-1)). Let alpha Omega. Then1. the set bB(alpha)' = {f is an element of bB' : f is Borel-alpha} is nowhere dense and perfect in (bB', d), and for any beta Omega, there exists D, a dense subset of (bB(alpha)', d) such that f(-1) is an element of bB(beta), for any f is an element of D,2. the set bH(alpha) = {f is an element of bH : f is Borel-alpha} is nowhere dense and perfect in (bH, d), and for any beta Omega a successor ordinal, there exists D, a dense subset of (bH(alpha+1), d) such that f(-1) is an element of bH beta, for any f is an element of D, and3. the set (bH(alpha), d*) is closed and nowhere dense in (bH, d*), and for any beta Omega, the set H(alpha, = beta) = {f is an element of bH(alpha) : f(-1) is an element of bH(beta)) is closed and nowhere dense in (bH, d*). (C) 2020 Elsevier B.V. All rights reserved.