TRANSFORMING LEBESGUE-STIELTJES INTEGRALS INTO LEBESGUE INTEGRALS

Abstract

A recent theorem of the author for continuous functions of bounded variation is extended to include the discontinuous case. Given $h$ of bounded variation on a closed interval $K$ let $s(t, y)$ be the total number 0, 1, 2 of the following conditions which hold at $(t, y) ϵ K × ℝ$:

• $y = h(t)$,
• $y$ lies strictly between $h(t–)$ and $h(t)$,
• $y$ lies strictly between $h(t)$ and $h(t+)$.

Given $f$ Lebesgue-Stieltjes integrable against $d$ h we can define $\bar f$ almost everywhere on $ℝ$ by $\bar{f}(y) = ∑_{tϵK}f(t) s(t, y)$ where the nonzero terms form a finite sum. The function $\bar f$ is Lebesgue integrable and its integral $\int_{ - \infty }^\infty {\bar f} \left( y \right)dy = \int_\kappa f \left| {dh}\right|$. Among the special cases is a generalization of Banach's indicatrix theorem.