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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
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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)