LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Cantor basic intervals have their expected masses

Statement

Assume the Axiom of Countable Choice. For the middle-thirds Cantor set, a level- $m$ basic interval is

$I_{b} = [\sum_{j = 1}^{m} 2 b_{j} 3^{- j}, \sum_{j = 1}^{m} 2 b_{j} 3^{- j} + 3^{- m}], b_{j} \in {0, 1} .$

Every such interval has Cantor measure $μ_{c} (I_{b}) = 2^{- m}$ , including $m = 0$ , where the interval is $[0, 1]$ .

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, the Cantor measure is the Lebesgue–Stieltjes measure of the Cantor function extended by zero to the left and one to the right. The Cantor measure

[F2]

The Cantor measure is atomless, is a probability, and is concentrated on $C$ , under Countable Choice. The Cantor measure is a singular atomless probability measure concentrated on the Cantor set

[F3]

The Cantor function equals its digit-defined function $γ$ on $C$ . The Cantor function is well defined, satisfies $c (x) \leq c (y)$ whenever $x \leq y$ , is surjective onto $[0, 1]$ , and is constant on every interval removed from the Cantor set

[F4]

Under the standing Countable Choice hypothesis, for continuous distribution functions the interval formula gives $μ_{F} ([a, b]) = F (b) - F (a^{-})$ , and $μ_{F} ({a}) = F (a) - F (a^{-})$ . Interval formulas and atoms for a Lebesgue-Stieltjes measure

[F5]

On $C$ , $γ$ halves each ternary digit and sums the resulting binary digits. The Cantor function on $[0, 1]$ , defined on the Cantor set through ternary digits and extended constantly across each removed interval

Proof

1.1

The left endpoint has digits $2 b_{1}, \dots, 2 b_{m}$ followed by zeros; the right endpoint has that same prefix followed by twos, since $\sum_{j > m} 2 3^{- j} = 3^{- m}$ . Their Cantor-function values are respectively $q = \sum_{j = 1}^{m} b_{j} 2^{- j}$ and $q + \sum_{j > m} 2^{- j} = q + 2^{- m}$ .

F3F5

2.1

The distribution-function interval formula and absence of endpoint atoms give $μ_{c} (I_{b}) = (q + 2^{- m}) - q = 2^{- m}$ . This also applies at the endpoints zero and one of the extended distribution function; for $m = 0$ it agrees with probability mass one.

F1F2F4step 1.1∎

Depends on

Used by

The sharp interval bound for Cantor measure Lemma

Dependency tree · two levels

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Sources

Fremlin 264J(a,d); Bishop–Peres Example 1.4.2 (uniform digits) (standard reference, not scraped)