Does a non-measurable set always contain a bounded non-measurable subset?

Question

This question might be easy but I lack the proper intuition of non-measurable sets.

The only example of a non-measurable set I can think of is the Vitali set.

Answer 1

I will write down explicitly what is meant in @Teresa Lisbon's comment.

Let $S$ be an unbounded non-measurable set. For $n\ge 1$ call $$S_n=S \cap [-n; n]$$ Argue by contradiction that all the $S_n$ are measurable. Then their union is measurable. However $$\bigcup_{n=1}^{\infty} S_n = \bigcup_{n=1}^{\infty} S \cap [-n, n] = S \cap \bigcup_{n=1}^{\infty} [-n, n] = S \cap \Bbb R = S$$ Hence $S$ is measaurable, contradicting our assumptions.

This means that some $S_n$ is not measurable. And clearly $S_n \subseteq S$ is bounded.