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The Cantor set is an uncountable subset of of Lebesgue measure zero
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). The Cantor middle-thirds set (The Cantor middle-thirds set as the intersection of the sets obtained by removing open middle thirds) is Lebesgue measurable with
and is uncountable (Finite, countably infinite, countable, uncountable).
Facts & Assumptions
Given: The Axiom of Countable Choice and the Cantor middle-thirds set .
Assuming countable choice, if and only if has measure zero in the covering sense (A subset of has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers, Measure zero (a countable cover by intervals of total length below every ) and content zero (a finite such cover)).
Assuming countable choice, every with is Lebesgue measurable with (Assuming countable choice, is a sigma-algebra containing every elementary set and is a complete measure extending elementary volume).
has content zero, and therefore measure zero, and is uncountable (The Cantor set is compact, perfect, uncountable, nowhere dense and of measure zero, and it contains no interval of positive length, so its only nonempty connected subsets are single points, claims 2 and 4; The Cantor middle-thirds set as the intersection of the sets obtained by removing open middle thirds, Finite, countably infinite, countable, uncountable).
Proof
The published theorem gives that has measure zero in the covering sense, so the agreement theorem gives .
A set of Lebesgue outer measure zero is Lebesgue measurable with measure zero, so , while the same published theorem gives that is uncountable.
Depends on
- A subset of $\mathbb{R}$ has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers
- The Cantor set is compact, perfect, uncountable, nowhere dense and of measure zero, and it contains no interval of positive length, so its only nonempty connected subsets are single points
- The Cantor middle-thirds set as the intersection of the sets $C_n$ obtained by removing open middle thirds
- Finite, countably infinite, countable, uncountable
- Assuming countable choice, $\mathcal{L}(\mathbb{R}^n)$ is a sigma-algebra containing every elementary set and $\lambda_n$ is a complete measure extending elementary volume
- Measure zero (a countable cover by intervals of total length below every $\varepsilon$) and content zero (a finite such cover)
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
- A continuous image of a Lebesgue measurable subset of ℝ can be nonmeasurable Corollary
- A continuous preimage of a Lebesgue measurable subset of ℝ can be nonmeasurable Corollary
- There is a Lebesgue measurable subset of ℝ that is not Borel Corollary
- Cantor function has singular distributional derivative Counterexample
- Equality almost everywhere with a measurable function can fail on an incomplete space Counterexample
- Two Radon-Nikodym derivatives can differ on a null set Counterexample
- The ACL and Sobolev analytic definition of quasiconformality Definition
- The Cantor function has derivative 0 almost everywhere, is not differentiable on the Cantor set, and still rises from 0 to 1 Example
- The complement of the Cantor set in [0,1] has Lebesgue measure one, computed from the removed intervals Example
- Assuming the Axiom of Choice, every Lebesgue measurable subset of ℝ is a Borel set False statement
- FALSE: a monotone function has at most countably many points of non-differentiability False statement
- FALSE: composing a Lebesgue measurable function with a continuous map preserves measurability False statement
- FALSE: equality almost everywhere with a measurable function implies measurability False statement
- FALSE: every Riemann integrable function on a closed bounded interval is Borel measurable False statement
- FALSE: the Radon-Nikodym derivative is a uniquely determined function False statement
- The homeomorphism x ↦ x + c(x) sends the Cantor set onto a compact set of Lebesgue measure 1 Lemma
- The Cantor measure is a singular atomless probability measure concentrated on the Cantor set Proposition
Dependency tree · two levels
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Sources
- John K. Hunter, Measure Theory (UC Davis lecture notes), Example 2.14 (standard reference, not scraped)
- T. Tao, An Introduction to Measure Theory (GSM 126), Exercise 1.2.9 (standard reference, not scraped)