Does a Set with Empty Interior Have Measure Zero?

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Discussion Overview

The discussion revolves around the question of whether a set with empty interior necessarily has measure zero. Participants explore examples and counterexamples, particularly focusing on the Cantor set and the set of irrational numbers.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that a set with empty interior has measure zero.
  • Another participant points out that the Cantor set has an empty interior.
  • A different participant notes that the Cantor set is known to have measure zero, implying it supports the initial claim.
  • One participant expresses uncertainty about their earlier assertion, indicating a potential misunderstanding.
  • A participant clarifies that while the standard middle thirds Cantor set has measure zero, not all Cantor sets necessarily do, suggesting a distinction that may challenge the initial claim.
  • Another participant introduces the set of irrational numbers between 0 and 1 as a counterexample, questioning both its interior and measure, and dismisses reliance on textbooks for this reasoning.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the relationship between empty interior and measure zero.

Contextual Notes

There are limitations in the discussion regarding the definitions of measure and the types of sets being considered, as well as the potential for misunderstanding the properties of different Cantor sets.

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does a set with empty interior have measure zero? I think it does...
 
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Doesn't the Cantor set have empty interior?
 
ya and I read somewhere that it has measure zero. So this wouldn't be a counterexample.
 
ok i may be wrong on this...
 
The Cantor set does have an empty interior and also has measure zero.

I never worked with measurable spaces yet, but if I had to guess, I would say a set with empty interior does in fact have measure zero.
 
no actually it says in my book that the standard middle thirds cantor set has measure zero, but not any cantor set. So I guess its incorrect.
 
come on guys, look at the set of irrational numbers between 0 and 1. what is the interior? what is the measure?

take your book and put it under your coffee cup while you think about this.
 

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