Local times for functions with finite variation: two versions of Stieltjes change of variables formula

We introduce two natural notions for the occupation measure of a function V with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show that both measures are absolutely continuous with respect to the Lebesgue measure. Occupation densities can be thought of as local times of V , and are described by a Meyer–Tanaka like formula. As an example of applications, we derive an extension of Banach’s indicatrix theorem.