A Stieltjes Type Extension of the $L^{r}$-Perron Integral

Abstract

We explore properties of $L^{r}$-derivates with respect to a monotone increasing Lipschitz function. We then define $L^{r}$-ex-major and $L^{r}$-ex-minor functions with respect to a monotone increasing Lipschitz function and use these to define a Perron-Stieltjes-type integral which extends the integral of L. Gordon.