This question is a mixture of several related problems.

  1. Is there any concrete examples (the examples that are easy to construct and think about, and of which the construction doesn't include heavy language of set theory) for sets that are not Lebesgue measurable?
  2. Without Axiom of Choice, is the existence of sets that are not Lebesgue measurable unprovable? If so, how to prove this unprovability?
  3. Is there concrete, simple examples of sets that are Lebesgue measurable and are not Borel measurable?
  4. Is the class of all Lebesgue Measurable sets a maximal collection of subsets of $\mathbb{R}$ on which we can define a nonzero measure?

closed as too broad by Stella Biderman, Hans Lundmark, Stefan Mesken, Did, TheSimpliFire Feb 3 '18 at 9:05

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