On the Substitution Rule for Lebesgue–stieltjes Integrals

We show how two change-of-variables formulæ for Lebesgue–Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of “mass splitting phenomenon” arises. Let M : [a, b]→ R be increasing. Then the measure corresponding to M may be defined to be the unique Borel measure μ on [a, b] such that for each continuous function f : [a, b] → R, the integral ∫ [a,b] f dμ is equal to the usual Riemann-Stieltjes integral ∫ b a f(x) dM(x). Now let f : [a, b]→ R be a bounded Borel function. Then by definition, the Lebesgue-Stieltjes integral ∫ b a f(x) dM(x) is equal to ∫ [a,b] f dμ. If a < c < b, then of course the equation ∫ b