A set of measure zero in x-y plane

Question

A set of measure zero in x-y plane has the property that almost every line parallel to the y-axis intersects it in a set of linear measure zero. What does this mean? How to write such statement in the form mathematical symbols ? And How to prove ?

Answer 1

What about the set $\mathbb{Q}\times\mathbb{Q}$ ? For any line $x\times\mathbb{R}$, $$x\times\mathbb{R}\cap\mathbb{Q}\times\mathbb{Q}=x\times\mathbb{Q}$$ if $x\in\mathbb{Q}$, which is countable; and the intersection is zero otherwise.