I am studying for my midterm review. Could someone give me hints on how to start this question?
Useful theorem: Vitali Any set of real numbers with positive outer measure contains a subset that fails to be measurable.
Thanks for the help!
2026-03-29 14:57:54.1774796274
Prove that there are uncountably many non-measurable sets.
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To be honest, I'm not sure how to make use of the theorem - there may be a way to apply it, but it looks harder than a much more straightforward approach.
Three facts are important: (1) There is a non-measurable set. (2) All countable sets are measurable. (3) The union of measurable sets is measurable. [If you don't have all three of these yet, then this approach might not be workable for you.]
So, for example: take any non-measurable set $A$. What happens if you take away one point?