Riemann–Stieltjes integral

Riemann–Stieltjes Integral

The Riemann–Stieltjes integral is a generalization of the Riemann integral, introduced independently by Bernhard Riemann and Thomas Joannes Stieltjes. It serves to extend the concept of integration with respect to a variable to integration with respect to a function. This extension is particularly useful in the fields of probability theory, functional analysis, and differential equations, where it provides a robust framework for integrating functions against measures and distributions.

Definition

Given two functions \(f\) and \(g\) defined on a closed interval \([a, b]\), where \(f\) is continuous and \(g\) is of bounded variation, the Riemann–Stieltjes integral of \(f\) with respect to \(g\) over \([a, b]\) is denoted by \(\int_a^b f(x) \, dg(x)\) and is defined as the limit of a sum, if this limit exists. Specifically, for any partition \(P = \{a = x_0 < x_1 < \ldots < x_n = b\}\) of the interval \([a, b]\), and for any choice of points \(c_i \in [x_{i-1}, x_i]\), the integral is the limit of the sum \[ \sum_{i=1}^n f(c_i) \left( g(x_i) - g(x_{i-1}) \right) \] as the norm of the partition \(P\) (the length of the largest subinterval) approaches zero, provided this limit exists and is the same for all such choices of points \(c_i\).

Properties

The Riemann–Stieltjes integral shares many properties with the Riemann integral, including linearity and additivity. However, it also exhibits unique characteristics due to the involvement of the function \(g\). For instance, if \(g\) is differentiable and its derivative \(g'\) is continuous on \([a, b]\), then the Riemann–Stieltjes integral of \(f\) with respect to \(g\) can be computed as a Riemann integral: \[ \int_a^b f(x) \, dg(x) = \int_a^b f(x) g'(x) \, dx. \]

Applications

The Riemann–Stieltjes integral finds applications in various areas of mathematics and its applications. In probability theory, it is used to define the expectation of continuous random variables. In functional analysis, it plays a role in the study of linear operators. Moreover, it is utilized in the formulation of the Stieltjes moment problem and in the theory of differential equations for integrating functions against measures.

Examples

1. If \(g(x) = x\), the Riemann–Stieltjes integral reduces to the standard Riemann integral. 2. If \(f(x) = x\) and \(g(x) = \lfloor x \rfloor\), the integral \(\int_0^b x \, d\lfloor x \rfloor\) computes the sum of the first \(b-1\) positive integers.

See Also

• Lebesgue integral
• Measure theory
• Stieltjes moment problem

References

• Rudin, Walter. Principles of Mathematical Analysis. McGraw-Hill, 1976.
• Apostol, Tom M. Mathematical Analysis. Addison-Wesley, 1974.