Integrator function (Riemann–Stieltjes)

The function α whose increments define the weights in Riemann–Stieltjes sums.

Integrator function (Riemann–Stieltjes)

In the Riemann–Stieltjes integral $\int_a^b f\,d\alpha$ , the function

\alpha:[a,b]\to\mathbb{R}

is called the integrator (or integrator function). For a partition $P:a=x_0<\cdots<x_n=b$ , the weights are the increments

\Delta\alpha_i=\alpha(x_i)-\alpha(x_{i-1}).

Typically one assumes $\alpha$ is increasing (or more generally of bounded variation) to ensure good behavior of the integral and to guarantee that upper/lower sum definitions make sense.

Examples:

For the usual Riemann integral, the integrator is $\alpha(x)=x$ .
If $\alpha(x)=x^2$ (increasing on $[0,\infty)$ ), then $\int_a^b f\,d\alpha$ weights subintervals according to changes in $x^2$ .
If $\alpha$ has jumps, the integral can encode discrete contributions (a classical motivation for Stieltjes integration).