Note on the First Fundamental Theorem for Riemann-Stieltjes Integrals

Riemann-Stieltjes integrals

This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltj...

The Fundamental Theorem on Quadratic First Integrals.

4: Riemann Sums, Riemann Integrals, Fundamental Theorem of Calculus

Proposition 1.3. Let X be a subset of R, let x0 be a limit point of X, let f : X → R be a function, and let L be a real number. Then the following two statements are equivalent. • f is differentiable at x0 on X with derivative L. • For every ε > 0, there exists a δ = δ(ε) > 0 such that, if x ∈ X satisfies |x− x0| < δ, then |f(x)− [f(x0) + L(x− x0)]| ≤ ε |x− x0| . Corollary 1.4 (Mean Value Theor...

Cauchy-stieltjes and Riemann-stielt Jes Integrals

Two Point Gauss–legendre Quadrature Rule for Riemann–stieltjes Integrals

In order to approximate the Riemann–Stieltjes integral ∫ b a f (t) dg (t) by 2–point Gaussian quadrature rule, we introduce the quadrature rule ∫ 1 −1 f (t) dg (t) ≈ Af ( − √ 3 3 ) + Bf (√ 3 3 ) , for suitable choice of A and B. Error estimates for this approximation under various assumptions for the functions involved are provided as well.