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A Lebesgue measurable subset of with empty interior has measure zero
Statement
Assume the Axiom of Countable Choice. Every Lebesgue measurable subset of with empty interior has measure zero.
Facts & Assumptions
Given: The Axiom of Countable Choice and the Smith-Volterra-Cantor set .
is nowhere dense and does not have measure zero (The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero).
is nowhere dense when the interior of its closure is empty (Nowhere dense, meager (first category), residual, and second category subsets of ).
A subset of has Lebesgue outer measure zero if and only if it is null in the sense of countable closed-interval covers (A subset of has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers).
Refutation
The Smith-Volterra-Cantor set is nowhere dense by [L1], so [F1] gives that it has empty interior.
The same source item [L1] says that is not null, so [L2] gives ; since Lebesgue measure is nonnegative, this means .
So a measurable set can have empty interior and still have positive Lebesgue measure; the Smith-Volterra-Cantor set refutes the statement.
Depends on
- The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero
- Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$
- 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 Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
46 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 (UC Davis lecture notes), Chapter 2 (standard reference, not scraped)
- A. Jin, Cantor sets in topology, analysis, and financial markets (standard reference, not scraped)