On Borel measures on separable metric spaces (Q1308859)

The author generalizes a known result, viz., the existence of a Lebesgue measurable set \(E\subset [0,1]\) such that for any open set \(U\subset (0,1)\), \(m(U\cap E)> 0\) and \(m(U\backslash E)> 0\), where \(m\) is the Lebesgue measure. In doing so, the author proves the following Theorem: Let \(m\) be a positive Borel measure on a separable metric space \(X\) such that \(m(X)= 1\), and \(m\) vanishes on every countable set. Then there is an \(F_ \sigma\)-set \(E\) such that for any open set \(U\) with \(m(U)>0\), we have \(m(U\cap E)>0\) and \(m(U\backslash E)>0\).