σ-FINITE BOREL MEASURES ON THE REAL LINE
Thomson · Real Analysis Exchange · 1997
A characterization is given of those Borel measures on the real line that can be expressed as the total variation measure of an ACG * function.Let µ be a measure defined on the Borel subsets of an interval [a, b].If µ is absolutely continuous with respect to Lebesgue measure (that is, if µ(N ) = 0 for every Borel set N of Lebesgue measure zero) and if µ ([a, b]) < ∞ then µ can be represented in the formwhere f is absolutely continuous on [a, b] and µ f is the corresponding Lebesgue-Stieltjes measure.Beginning students of analysis learn this material routinely.It seems, though, that there has been little discussion of the σ-finite case.If µ is not finite, but is σ-finite, is there a representation similar to this available?Part of such a representation is immediately available from the Radon-Nikodym theorem and a theorem of Lusin.Any absolutely continuous, σ-finite measure µ on [a, b] can be represented asfor some measurable, finite a.e.function g.But Lusin's theorem (eg., see [1, p. 113]) asserts the existence of a continuous function f with f = g a.e.This gives