# Riemann-Stieltjes Integral for discontinuous integrator

I was wondering if there is a good general technique for evaluating the Riemann-Stieltjes integral in cases where the integrator is discontinuous on a few points. For example: find $$\int_{0}^{2} f(x) d\alpha(x)$$ where $$f(x) = e^{2x}$$ and $$\alpha = x + 1$$ for $$x$$ in $$[0, 1]$$ and $$\alpha = 6x$$ for $$x$$ in $$(1, 2]$$. (Jumps from 2 to 6.)

It should be Riemann integrable based on the Riemann condition. Then I would like to split it apart and then use the reduction to a Riemann integral in terms of $$\alpha'(x)$$, but the condition for this is that $$\alpha'(x)$$ is continuous, but the derivative does not even exist for the interval $$[0, 2]$$. So instead I thought of splitting the integral apart into $$\int_{0}^{1}f(x)d\alpha(x) + \int_{1}^{2}f(x)d\alpha(x)$$, where the first term exists. Then for the second we might take $$\int_{1}^{1 + \epsilon} f(x)d\alpha(x) + \int_{1 + \epsilon}^{2}f(x)d\alpha(x)$$. But I am not sure that taking the limit here as $$\epsilon \to 0$$ is relevant, since the integral as a function of the endpoints might not be continuous.

Your example is $$\alpha(dx)=4\delta _1(dx)+g(x)dx$$ where $$g(x)=1$$ on $$[0,1]$$ and $$6$$ on $$(1,2]$$ meaning that if $$f$$ is continuous on $$[0,2]$$ we have $$\int_{[0,2]}fd\alpha=4f(1)+\int_0^2f(x)g(x)dx$$ Discontinuities of the increasing function $$\alpha$$ correspond to Dirac masses of the corresponding positive measure $$d\alpha(x)$$ also denoted $$\alpha(dx)$$ in measure theory. It is a pity that many classes on measure theory do not stress the case of a measure on $$R$$ or an interval, essentially described by an increasing function.
• Your integral is of the Lebesgue-Stieltjes type. As Martin R mentioned earlier, the Riemann-Stiletjes integral may not exists when the integrand $f$ and the integrator $g$ have common discontinuity of the same type (right or left). This is not the case with Lebesgue-Stieltjes since the integrator $g$ is right-continuous with left limits and becomes a (finite) measure. Jun 1, 2023 at 18:03