Example of a subset of $\mathbb{R}^2$ which is not $\sigma$-finite under Lebesgue measure?

Question

We know open and closed sets are $\sigma$-finite. What would be the example of a subset of $\mathbb{R}^2$ which is not $\sigma$-finite under Lebesgue measure?

Answer 1

(Thanks to @zhw pointing out my mistake) If we allow $E\subset\mathbb{R}^2$ to be a non-measurable set, then obviously it cannot be a countable union of measurable sets of a finite measure. We can construct a non-measurable subset of $[0,1]^2$ in a similar way constructing a Vitali set. This can be an example of a non-$\sigma$-finite set that you are looking for. However, we cannot find any examples among measurable sets. The reason is as follows. Note that every measurable set $E$ can be expressed as $$ E=\bigcup_{n=1}^\infty E_n $$ where $E_n =E\cap [-n,n]^2$ and that each $E_n$ has a finite measure $m(E_n)\le m([-n,n]^2)= 4n^2$. This shows that there are no measurable, non-$\sigma$-finite subset of $\mathbb{R}^2$.