On Lebesgue measure in 3 dimensional euclidean space

Question

Suppose $E_{\alpha}=[(x,y,z):x^2+y^2+z^2=\alpha^2]$, $\alpha$ is irrational. Define $E=\bigcup E_{\alpha}$, where $\alpha$ varies over all irrationals, what is lebesgue measure of $E$, and what is the lebesgue measure of any open set containing $E^c$?

Answer 1

$E^{c}$ has measure $0$ because it is a countable union of sets of measure $0$. This implies that measure of $E$ is $\infty $. The measure of an open set containing $E^{c}$ can be finite or infinity.