Riemann–Stieltjes integral

Summary

The Riemann–Stieltjes integral $\int_a^b f \, dg$ integrates a function $f$ against another function $g$, the integrator. Its sums weight each value of $f$ by the change of $g$ over a subinterval instead of by the length of the subinterval. With $g(x) = x$ it is the Riemann integral. It handles sums and integrals in one formula. It is the setting of the Riesz representation of continuous linear functionals on $C[a, b]$, and of expectations written with a cumulative distribution function.