Riemann-Stieltjes Integral

Core Idea

The Riemann-Stieltjes integral is a generalization of the standard Riemann integral. The fundamental change is replacing the uniform measure of length, $d x$ , with a more general, non-uniform measure of "weight," $d g (x)$ , provided by an integrator function $g (x)$ . It would correspond to change the Lebesgue measure in $R$ .

This allows integration to account for non-uniform distributions, discrete jumps, and other scenarios that the standard Riemann integral cannot handle.

Mathematical Formulation

The standard Riemann integral is the limit of a sum where each function value is weighted by the length of its subinterval, $Δ x_{i}$ .

\int_{a}^{b} f (x) d x = lim_{| P | \to 0} \sum_{i = 1}^{n} f (c_{i}) Δ x_{i} where Δ x_{i} = x_{i} - x_{i - 1}

The Riemann-Stieltjes integral generalizes this by weighting each function value by the change in the integrator function $g (x)$ over the subinterval, $Δ g_{i}$ .

\int_{a}^{b} f (x) d g (x) = lim_{| P | \to 0} \sum_{i = 1}^{n} f (c_{i}) Δ g_{i} where Δ g_{i} = g (x_{i}) - g (x_{i - 1})

Key Insights

The integrator function $g (x)$ determines the "importance" of different parts of the integration interval.

Non-Uniform Weighting: If $g (x)$ changes rapidly in a region, that region is given more weight. If $g (x)$ is constant, that region contributes nothing to the integral.
Discrete Jumps: If $g (x)$ is discontinuous and has a jump at a point, the integral can incorporate the value of $f (x)$ at that specific point, weighted by the size of the jump.

Practical Example: Representing a Sum

The Riemann-Stieltjes integral can represent a discrete sum. To represent the sum $\sum_{i = 1}^{\infty} f (i)$ , we use the floor function $g (x) = ⌊ x ⌋$ as the integrator.
The floor function is a step function that increases by exactly 1 at each positive integer. The integral therefore only accumulates values at these integer points.

\sum_{i = 1}^{\infty} f (i) = \int_{0.5}^{\infty} f (x) d ⌊ x ⌋

This elegantly unifies the concepts of discrete summation and continuous integration.