The measure of $([0,1]\cap \mathbb{Q})×([0,1]\cap\mathbb{Q})$ We know that $[0,1]\cap \mathbb{Q}$ is a dense subset of $[0,1]$ and has measure zero, but what about $([0,1]\cap \mathbb{Q})\times([0,1]\cap \mathbb{Q})$? Is it also a dense subset of $[0,1]\times[0,1]$ and has measure zero too?
Besides, what about its complement? Is it dense in $[0,1]\times[0,1]$ and has measure zero?

To give a somewhat comprehensive answer:


*

*the set in question is countable (as a product of countable sets), so it is of measure zero (because any countable set is zero with respect to any continuous measure, such as Lebesgue measure).

*it is also dense, because it is a product of dense sets.

*it has measure zero, so its complement has full measure.

*its complement has full measure with respect to Lebesgue measure, so it's dense in $[0,1]^2$

