##  [Stieltjes Integral](/stieltjes-integral-0) 

 Definition

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

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

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

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

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

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

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

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

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.