Riemann–Stieltjes integral

An integral defined using increments of an integrator function.

Riemann–Stieltjes integral

A Riemann–Stieltjes integral of a bounded function $f:[a,b]\to\mathbb R$ with respect to an integrator function $\alpha:[a,b]\to\mathbb R$ is a number

\int_a^b f\,d\alpha

such that for every $\varepsilon>0$ there exists $\delta>0$ with the property that, for every tagged partition $(P,\{t_i\})$ with mesh $\|P\|<\delta$ ,

\left|\sum_{i=1}^n f(t_i)\bigl(\alpha(x_i)-\alpha(x_{i-1})\bigr)-\int_a^b f\,d\alpha\right|<\varepsilon.

(The sum is called a Riemann–Stieltjes sum.)

This generalizes the Riemann integral (take $\alpha(x)=x$ ) and is especially well-behaved when $\alpha$ is a function of bounded variation ; see Riemann–Stieltjes integrability and integration by parts .

Examples:

If $\alpha(x)=x$ , then $\int_a^b f\,d\alpha=\int_a^b f(x)\,dx$ .
If $\alpha$ is constant on $[a,b]$ , then $\int_a^b f\,d\alpha=0$ (whenever the integral is defined).