How Many Non-Measurable Sets Exist?

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SUMMARY

The discussion centers on the existence of non-measurable sets, highlighting examples from mathematical literature. The Vitali set, which involves addition modulo an irrational number and equivalence classes, is a well-known case. Edward Van Vleck's construction of two subsets of (0,1) demonstrates that either one has measure 1 while the other has measure 0, or both possess upper measure 1, rendering them non-measurable. Additionally, the Bernstein set is mentioned as a notable example, constructed via transfinite induction to intersect every uncountable closed set.

PREREQUISITES
  • Understanding of measure theory concepts
  • Familiarity with irrational numbers and their properties
  • Knowledge of transfinite induction techniques
  • Basic comprehension of equivalence classes in set theory
NEXT STEPS
  • Research the properties of the Vitali set in detail
  • Explore Edward Van Vleck's construction of non-measurable sets
  • Study the implications of the Bernstein set in topology
  • Learn about the Axiom of Choice and its role in measure theory
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Mathematicians, students of advanced mathematics, and anyone interested in set theory and measure theory will benefit from this discussion.

fourier jr
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are there lots of examples of non-measurable sets? the one that seems to be in most textbooks involves a type of addition mod an irrational number with equivalence classes, etc etc, which in some books is done geometrically as rotations of a circle through an irrational angle. that example was by vitali, but there is another one by edward van vleck where he constructs two subsets of (0,1) in a way that either the measure of one is 1 & the other has measure 0, or they both have what he called "upper measure" =1, & therefore they're both non-measurable. how many other ones are out there that I don't know about?
 
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My favorite is due to Bernstein.

You construct (by transfinite induction) a set B so that both B and its complement meet every uncountable closed set.
 

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