1) Everywhere one can find many things about Lebesgue measure on $\mathbb{R}^2$. But what about arbitary measure? Is there a non-zero measure on $\mathbb{R}^2$ such that all subsets of $\mathbb{R}^2$ are measurable?
2) Another variant of the question:
Is there a measure on $\mathbb{R}^2$ such that the measure of a unit square is one, the measure is invariant under isometries and all subsets of $\mathbb{R}^2$ are measurable?
EDITED: I miss one important condition that a mesure of any bounded set should be finite.