Area of infinite number of points

Question

Let's say I have a set $S$ of points in $\mathbb{R} ^2$, and I was able to find a bijection from $S$ to the unit disk. Can I say that the area of points in $S$ is $\pi$? I doubt it.. so my alternative question is, what additional constraints on $S$ do I need in order to prove that?

Answer 1

Just a bijection? If so, you can at best say that $S$ has the cardinality of $\mathbb{R}$. If you have more (continuous), you can say more.

Answer 2

Of course not: consider the function from the unit disk $\{(x,y)\in\mathbb R^2\mid x^2+y^2\leq1\}$ to the disk of radius $2$, $\{(x,y)\in\mathbb R^2\mid x^2+y^2\leq 4\}$, given by $(x,y)\mapsto(2x,2y)$. Geometrically this magnifies the unit disk by a factor of $2$, which is evidently a bijection, but doesn't conserve the area. If you require that the map is an isometry (i.e. it preserves distances), then it is true that area is conserved.