On Absolute Continuity

Abstract

We prove that in any dimension a variational measure associated with an additive continuous function is σ-finite whenever it is absolutely continuous. The one-dimensional version of our result was obtained in [1] by a different technique. As an application, we establish a simple and transparent relationship between the Lebesgue integral and the generalized Riemann integral defined in [7, Chap. 12]. In the process, we obtain a result (Theorem 4.1) involving Hausdorff measures and Baire category, which is of independent interest. As variations defined by BV sets coincide with those defined by figures [8], we restrict our attention to figures. The set of all real numbers is denoted by , and the ambient space of this paper is m where m ≥ 1 is a fixed integer. In m we use exclusively the metric induced by the maximum norm Ž · Ž. The usual inner product of x; y ∈ m is denoted by x · y, and 0 denotes the zero vector of . For an x ∈ m and e > 0, we let Bex‘ = { y ∈ m x Žx− yŽ < e}: