Riemann-Stieltjes Integral

Source: Principles of Mathematical Analysis

Riemann Integral

Let be a bounded real function defined on . A partition of is a finite set of points

Write . The upper and lower Riemann sums are given by

The upper and lower Riemann integrals are the extrema of these values

When these integrals are equal, is Riemann integrable on , and . The common value of these integrals is written as

Riemann-Stieltjes Integral

Let be monotonically increasing on . Given a partition of , write . We define the analogous sums and integrals with the same and from before.

Once again, when these are equal, is integrable with respect to , and . Note that setting recovers the original Riemann integral. We denote the common value of this integral as