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 $T_2\colon\A\to\A$.
Publié le : 1999-05-15
Classification:
simultaneous extension operator,
approximately continuous functions,
density topology,
54C20,
46E15,
54A10
@article{1231187601,
author = {Kol\'a\v r, Jan},
title = {Simultaneous Extension Operators for the Density Topology},
journal = {Real Anal. Exchange},
volume = {25},
number = {1},
year = {1999},
pages = { 223-230},
language = {en},
url = {http://dml.mathdoc.fr/item/1231187601}
}
Kolář, Jan. Simultaneous Extension Operators for the Density Topology. Real Anal. Exchange, Tome 25 (1999) no. 1, pp. 223-230. http://gdmltest.u-ga.fr/item/1231187601/