Is there an example for a limited set that is not measurable?

Question

I asked myself whether there is an example for a subset $U \subset \mathbb{R}^n$ with the property that it is limited but not Lebesgue-measurbale? Do you have a hint?

Answer 1

Assuming that by "limited" you mean "bounded," note that Vitali sets - the usual first example of a non-measurable set - are nonmeasurable subsets of $[0,1]$ ...

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On the other hand, there are some known restrictions on what a non-measurable set could possibly be. For example:

• Every set of outer measure $0$ is measurable.

  • Indeed, it turns out that a set $X$ is measurable if and only if there is some $G_\delta$ set $Y$ such that the symmetric difference $X\triangle Y$ has measure zero; this follows from the regularity of Lebesgue measure.

• Every Borel set is measurable.

  • In fact, this can be pushed further: every analytic set is measurable. Under further set-theoretic hypotheses we can extend this to even broader collections of sets - e.g. under appropriate large cardinal hypotheses, every projective set is measurable. This sort of thing is one of the topics studied in descriptive set theory, with a main theme being that - especially under large cardinal hypotheses - "reasonably definable" sets of real numbers are tame in various senses (including measurability).

Answer 2

Suppose otherwise. Then every set $S$ of real numbers would be measurable, because$$S=\bigcup_{n\in\mathbb Z}\bigl(S\cap[n,n+1)\bigr)$$and each set $\bigl(S\cap[n,n+1)\bigr)$ is bounded.