Integrator function

An integrator function on $[a,b]$ is a function $\alpha:[a,b]\to\mathbb R$ used to weight increments in the definition of the Riemann–Stieltjes integral : the associated sums use the differences $\alpha(x_i)-\alpha(x_{i-1})$ along partitions.

In most standard existence theorems, $\alpha$ is assumed to be monotone or, more generally, a function of bounded variation , which guarantees good control of these increments.

Examples:

• $\alpha(x)=x$ recovers the usual Riemann integral from the Riemann–Stieltjes integral.
• Any step function $\alpha$ on $[a,b]$ (for instance, one with a single jump) is an integrator function commonly used to model point-mass contributions.