Lebesgue measure of a certain subset in $\mathbb{R}^2$

Question

What is Lebesgue measure of the set $\{(a,b) \in \mathbb{R}^2 \mid a-b \in\mathbb{Q}\}$ in $\mathbb{R}^2$ ? I am guessing this is exactly measure of the diagonal subset, but unable to say rigorously.

Answer 1

The section of this set by $a$ is $E_a=\{b: a-b \in \mathbb Q\}$ which is a contable set. Hence each section has measure $0$ and Fubini's Theorem tells you that the set has measure $0$.

[By Fubini's Theorem $m_2(E)=\int m_1(E_a) dm_1(a)$ where $m_1$ is Lebesgue measure on $\mathbb R$ and $m_2$ is Lebesgue measure on $\mathbb R^{2}$].

Answer 2

Rotate it through $45^\circ$ and it becomes $\sqrt 2(\Bbb Q\times\Bbb R)$, which has measure $0$.