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A continuously differentiable integrator reduces Stieltjes integration to ordinary integration
Statement
Let be Riemann integrable. Suppose is continuous on , differentiable on , and extends continuously to . Then is Riemann–Stieltjes integrable with respect to and
Facts & Assumptions
Given: A Riemann-integrable and an integrator with continuous derivative on the compact interval.
The mean value theorem represents every increment of as (The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with ).
The continuous function is uniformly continuous (Heine-Cantor in : a continuous real function on a compact subset of is uniformly continuous, proved -natively from sequential compactness).
The continuous function is Riemann integrable, so the product is Riemann integrable (A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion, If are integrable on then so are , , , and , and ).
Riemann tagged sums converge to the Darboux integral (The Darboux and Riemann definitions agree: a bounded on is Darboux integrable with integral if and only if for every real there is a real such that for every tagged partition of mesh below ).
Proof
A Riemann-integrable function is bounded; choose with . For each partition interval, [L1] gives such that . Hence [L1]
By [L2], the absolute value of the right side is at most , which tends to zero with the mesh. By [L3] and [L4], the second sum in step 1.1 tends to . Thus all Stieltjes sums have the same limit, proving both existence and the formula.
Depends on
- Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral
- The identity integrator recovers the Riemann integral
- The mean value theorem, as the case $g(x) = x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a < b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $f(b) - f(a) = f'(c)(b-a)$
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- Heine-Cantor in $\mathbb{R}$: a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous, proved $\mathbb{R}$-natively from sequential compactness
- The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$
- For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$
- A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion
- If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$
- The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ for every tagged partition of mesh below $\delta$
Used by
- ∫₀¹ x d(x²)=2/3 Example
- A nonlinear reparametrisation leaves a Stieltjes integral unchanged Example
- For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals Theorem
- The two FTC forms for a Riemann–Stieltjes integral with a C¹ integrator Theorem
Dependency tree · two levels
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Sources
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.17 (standard reference, not scraped)