Example of sets that are not measurable?

Asked Feb 02 '18 at 13:45

Active Feb 02 '18 at 14:09

Viewed 271 times

This question is a mixture of several related problems.

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?
Without Axiom of Choice, is the existence of sets that are not Lebesgue measurable unprovable? If so, how to prove this unprovability?
Is there concrete, simple examples of sets that are Lebesgue measurable and are not Borel measurable?
Is the class of all Lebesgue Measurable sets a maximal collection of subsets of $\mathbb{R}$ on which we can define a nonzero measure?

edited Feb 02 '18 at 14:09

asked Feb 02 '18 at 13:45

Ma Joad

6

Have you looked at the myriad of questions that essentially already asked these before? – Asaf Karagila Feb 02 '18 at 14:01
@AndrésE.Caicedo Could you please give proofs of the statements, or tell me where to find a proof for each? – Ma Joad Feb 02 '18 at 14:07
Asaf's suggestion is the way to go. – Andrés E. Caicedo Feb 02 '18 at 14:11
I don’t feel like this can be closed as a duplicate because I am not sure if there is a question that addresses all of these points, but please use the search bar at the top of the page, or search in google (add “math stack exchange” to the search term) – Stella Biderman Feb 02 '18 at 14:13
@AndrésE.Caicedo It hard to search because there are so many ways to say the same thing. I don't think Question 2 and 4 are too broad or have been asked in a similar way before. As for question 1 and 3, I have explained what I mean concrete. Many construction on internet are very set theoretical, which I don't want. – Ma Joad Feb 02 '18 at 14:17
@StellaBiderman: Note you can close as multiple-duplicates. – Asaf Karagila Feb 02 '18 at 14:17
@AndrésE.Caicedo: ok... clearing my comment... – Netchaiev Feb 02 '18 at 14:22
1
Probably NO for any notion of "concrete" you can think of. 2. Yes --- see Solovay's 1970 paper. 3. Yes --- see Becker's 1992 paper Descriptive set theoretic phenomena in analysis and topology and Non-Borel sets without axiom of choice. 4. See How much we can extend meaurable sets?

@DaveL.Renfro Thats very helpful – Ma Joad Feb 02 '18 at 22:21

0 Answers0