The Radon-Nikodým derivative in Euclidean spaces (Q752211)

Let \(\Phi\) be a \(\sigma\)-finite signed measure on the Borel \(\sigma\)- algebra of a Euclidean space and let m be Lebesgue measure. Three classical results - (1) inner regularity of \(\Phi\), (2) the existence of \(D\Phi\) and equality almost everywhere to the Radon-Nikodým derivative \(d\Phi\) /dm of the absolutely continuous part of \(\Phi\) and \((3)\quad D\Phi =0\) a.e. whenever \(\Phi\) is singular - known to be true whenever \(\Phi\) is finite on compact sets are shown to be false without this condition. Certain similar results are obtained in this more general setting: necessary and sufficient conditions for \(\Phi\) to be inner regular and analogues of (1) and (2) in terms of the lower derivate Ḏ\(\Phi\) whenever \(\Phi\) assumes the value \(\infty\) on a compact set.