On Absolute Continuity

✍ Scribed by Zoltán Buczolich; Washek F Pfeffer

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 209 KB
• Volume: 222
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5804

✦ Synopsis

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 • . The usual inner product of x y ∈ m is denoted by x • y, and 0 denotes the zero vector of m . For an x ∈ m and ε > 0, we let

* The results of this paper were presented to the Royal Belgian Academy on June 3, 1997.