A Functional Analytic Proof of the Lebesgue-Darst Decomposition Theorem

Abstract

The aim of this paper is to give a functional analytic proof of the Lebesgue--Darst decomposition theorem \cite{Darst}. We show that the decomposition of a nonnegative valued additive set function into absolutely continuous and singular parts with respect to another derives from the Riesz orthogonal decomposition theorem employed in a corresponding Hilbert space.