Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

D.2 RIEMANN–STIELTJES INTEGRAL

The Riemann–Stieltjes integral is a generalization of the Riemann integral where the subintervals of the Riemann sum in (D.3) are replaced as follows:

(D.13)

where h(x) is a nondecreasing function. In this way, different weightings can be assigned to the subintervals in the partition on the x-axis. This integration is performed with respect to the real function h(x); the original function g(x) is still called the integrand, and h(x) is known as the integrator. The Riemann–Stieltjes integral RS is defined in a manner similar to the Riemann integral R as follows: there exists for every such that

(D.14)

where . The following notation is used for the Riemann–Stieltjes integral on [a, b]:

(D.15)

If the function has negative parts, then the integral for each ...