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The Cantor function defines a nonclassical Stieltjes integrator and
Example
If is the Cantor function, then it is a continuous BV integrator and
Facts & Assumptions
Given: The Cantor function on .
The function is continuous and nondecreasing, with and (The Cantor function is continuous on , The Cantor function is well defined, satisfies whenever , is surjective onto , and is constant on every interval removed from the Cantor set).
A continuous integrand is integrable against a BV integrator (A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator).
Verification
Monotonicity and [L1] make BV, so existence follows from [L2]. Every tagged sum for the constant integrand telescopes: [L1, L2]
Thus its common limit is one. This computation invokes neither a derivative of nor measure theory.
Depends on
- The Cantor function is well defined, satisfies $c(x) \le c(y)$ whenever $x \le y$, is surjective onto $[0,1]$, and is constant on every interval removed from the Cantor set
- The Cantor function is continuous on $[0,1]$
- A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator
- Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral
- Laws of finite sums and finite products
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 131 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Example 3.3 (standard reference, not scraped)