The Density of Borel Sets

Abstract

Using a novel approach, we develop a result analogous to the Lebesgue density theorem for Borel sets using the notion of category: If $B\subset \lbrack 0,1]$ is a Borel set, then there exists a first category set $S\subset B$ with the property that for every $x\in B-S$ there exists $\varepsilon >0$ such that $B\cap (x-\varepsilon ,x+\varepsilon )$ is a residual subset of $(x-\varepsilon ,x+\varepsilon )$.