Does a Set with Empty Interior Have Measure Zero?

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SUMMARY

A set with empty interior does not necessarily have measure zero, as demonstrated by the discussion surrounding the Cantor set. The standard middle-thirds Cantor set is confirmed to have measure zero, while not all Cantor sets share this property. The conversation highlights the distinction between sets with empty interiors and their measure, particularly referencing the set of irrational numbers between 0 and 1, which complicates the relationship between interior and measure.

PREREQUISITES
  • Understanding of measure theory concepts
  • Familiarity with the Cantor set and its properties
  • Knowledge of measurable spaces
  • Basic comprehension of set theory
NEXT STEPS
  • Study the properties of the Cantor set in detail
  • Learn about measure theory and its applications
  • Explore the concept of interior points in topology
  • Investigate the measure of sets of irrational numbers
USEFUL FOR

Mathematicians, students of measure theory, and anyone interested in the properties of sets in topology and their measures.

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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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