The Cantor Set, Baire Category, and Measure Zero in $R$

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Nowhere dense, meager (first category), residual, and second category subsets of ℝ

Definition

F_σ and G_δ subsets of ℝ

Theorem

Baire category in ℝ, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so ℝ is not a countable union of nowhere dense sets

Corollary

A sequence of intervals covering [a,b] has total length at least b - a, so no interval of positive length has measure zero

Lemma

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

Definition

The Smith-Volterra-Cantor set: the same construction removing, at stage n ≥ 1, an open middle interval of length 4⁻ⁿ from each of the 2ⁿ⁻¹ remaining intervals

Theorem

The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero

Definition

The Cantor function on [0,1], defined on the Cantor set through ternary digits and extended constantly across each removed interval

Theorem

The Cantor function is well defined, satisfies c(x) ≤ c(y) whenever x ≤ y, is surjective onto [0,1], and is constant on every interval removed from the Cantor set

Remark

The Cantor Set, Baire Category, and Measure Zero in $R$

3 · Items

Nowhere dense, meager (first category), residual, and second category subsets of ℝ

F_σ and G_δ subsets of ℝ

Baire category in ℝ, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so ℝ is not a countable union of nowhere dense sets

ℚ is F_σ, meager and not G_δ, while the irrationals are G_δ, residual and not F_σ

Measure zero (a countable cover by intervals of total length below every ε) and content zero (a finite such cover)

If finitely many intervals cover a closed bounded interval [a,b], the sum of their lengths is at least b - a

A sequence of intervals covering [a,b] has total length at least b - a, so no interval of positive length has measure zero

Every at most countable subset of ℝ has measure zero

A countable union of measure-zero sets has measure zero, by countable choice

A set of content zero has measure zero

For a compact subset of ℝ, measure zero and content zero coincide

The Cantor middle-thirds set as the intersection of the sets Cₙ obtained by removing open middle thirds

The Cantor set is exactly the set of ∑_k ≥ 1 aₖ 3⁻ᵏ with every aₖ ∈ {0,2}, and this gives a bijection with {0,1}^ℕ

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 Smith-Volterra-Cantor set: the same construction removing, at stage n ≥ 1, an open middle interval of length 4⁻ⁿ from each of the 2ⁿ⁻¹ remaining intervals

The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero

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) ≤ c(y) whenever x ≤ y, is surjective onto [0,1], and is constant on every interval removed from the Cantor set

Why the nested-interval proof of Baire category in ℝ needs no choice

FALSE: every nowhere dense subset of ℝ has measure zero

FALSE: every subset of ℝ of measure zero is nowhere dense

FALSE: every set of measure zero has content zero

FALSE: ℚ is a G_δ subset of ℝ

FALSE: the Cantor set is countable because only countably many intervals were removed

The Cantor Set, Baire Category, and Measure Zero in RR

3 · Items

Nowhere dense, meager (first category), residual, and second category subsets of ℝ

F_σ and G_δ subsets of ℝ

Baire category in ℝ, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so ℝ is not a countable union of nowhere dense sets

ℚ is F_σ, meager and not G_δ, while the irrationals are G_δ, residual and not F_σ

Measure zero (a countable cover by intervals of total length below every ε) and content zero (a finite such cover)

If finitely many intervals cover a closed bounded interval [a,b], the sum of their lengths is at least b - a

A sequence of intervals covering [a,b] has total length at least b - a, so no interval of positive length has measure zero

Every at most countable subset of ℝ has measure zero

A countable union of measure-zero sets has measure zero, by countable choice

A set of content zero has measure zero

For a compact subset of ℝ, measure zero and content zero coincide

The Cantor middle-thirds set as the intersection of the sets Cₙ obtained by removing open middle thirds

The Cantor set is exactly the set of ∑_k ≥ 1 aₖ 3⁻ᵏ with every aₖ ∈ {0,2}, and this gives a bijection with {0,1}^ℕ

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 Smith-Volterra-Cantor set: the same construction removing, at stage n ≥ 1, an open middle interval of length 4⁻ⁿ from each of the 2ⁿ⁻¹ remaining intervals

The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero

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) ≤ c(y) whenever x ≤ y, is surjective onto [0,1], and is constant on every interval removed from the Cantor set

Why the nested-interval proof of Baire category in ℝ needs no choice

FALSE: every nowhere dense subset of ℝ has measure zero

FALSE: every subset of ℝ of measure zero is nowhere dense

FALSE: every set of measure zero has content zero

FALSE: ℚ is a G_δ subset of ℝ

FALSE: the Cantor set is countable because only countably many intervals were removed

The Cantor Set, Baire Category, and Measure Zero in $R$