Simultaneous Extension Operators for the Density Topology

Abstract

Given a set $M\subset\R$ of Lebesgue measure zero, let $\B_1\vert_M$ be the set of all restrictions to $M$ of bounded Baire one functions on $\R$ and $\A$ the set of all bounded approximately continuous functions on $\R$. We discuss the existence of simultaneous extension operators for $\B_1\vert_M$ and $\A$. We show that there exists a~positive linear operator $L\colon\B_1\vert_M\to\A$ such that $L(g)\vert_M=g$ for all $g\in\B_1\vert_M$, if and only if $M$ is a~scattered set and this is the case if and only if there exists a~continuous linear operator $L_1\colon\B_1\vert_M\to\A$ with the same property. Also, we show that there exist non-regular continuous linear operators $T_1\colon \ell_\infty \to \A$ and $T_2\colon\A\to\A$.