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The Cantor measure
Definition
Assume the Axiom of Countable Choice. Let be the Cantor function The Cantor function on , defined on the Cantor set through ternary digits and extended constantly across each removed interval. Define
By The Cantor function is well defined, satisfies whenever , is surjective onto , and is constant on every interval removed from the Cantor set and The Cantor function is continuous on , the function is nondecreasing and right-continuous on . The Cantor measure is the Lebesgue-Stieltjes measure
given by Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on .
Depends on
- The Cantor function on $[0,1]$, defined on the Cantor set through ternary digits and extended constantly across each removed interval
- 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 $[0,1]$
- Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $\mathbb{R}$
Used by
Dependency tree · two levels
48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John K. Hunter, Measure Theory, Example 2.37 (standard reference, not scraped)