On Absolute Continuity

We prove that in any dimension a variational measure associated with an additive continuous function is -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 2 m is denoted by x y, and 0 denotes the zero vector of m. For an x 2 m and " > 0, we let B"x‘ D y 2 m x Žx−yŽ < " : *The results of this paper were presented to the Royal Belgian Academy on June 3, 1997. †This author was supported by the Hungarian National Foundation of Scientific Research, Grant T019476 and FKFP 0189/1997.