An integral of a function f with respect to a function g, defined as the limit of sums Σ f(t_i)[g(x_{i+1}) - g(x_i)] as partitions refine; it generalizes the ordinary integral by allowing integration against an integrator function that may have jumps or singular behaviour.

Principle

Integration pairs function values with increments of an integrator g; when g has bounded variation the Stieltjes integral extends classical integration to include both continuous density contributions and discrete jumps.

Demonstration

If g is differentiable with derivative g', then ∫ f dg reduces to the ordinary integral ∫ f(x) g'(x) dx; if g is a cumulative jump function, the integral becomes a weighted sum of f at the jump points.

Misapplication

Assuming unrestricted interchangeability with measure-theoretic integrals without verifying conditions, or integrating against a highly oscillatory or nowhere‑of‑bounded‑variation g without suitable convergence theory.

Consequence

The construction links integrators and measures: a function of bounded variation defines a signed measure and the Stieltjes integral recovers integrals against that measure, useful in probability when g is a distribution function.

Reversal

The ordinary Riemann integral is the special case with g(x) = x; conversely, the Stieltjes integral permits more general integrators, including discontinuous cumulative functions.

Boundary

Existence for all continuous f is guaranteed when g has bounded variation; for more singular integrators one must invoke generalized integration theories (Young, Henstock–Kurzweil, or full measure-theoretic frameworks).

Semantic Tension

Compared to the Lebesgue integral, the Stieltjes integral emphasizes integration with respect to an integrator function rather than a fixed reference measure, producing different requirements on regularity and convergence.

Synthesis

The Stieltjes integral generalizes classical integration by summing function values against increments of an integrator function, unifying treatment of continuous densities and discrete jumps in a single framework.