$M:=\{(x,y,z)\in R^3:x^2+y^2\le z, z\le1\}$ is a paraboloid in $R^3$
I need to prove that M is Borel measurable. Do I need to do something like this? Because I am not really sure how solve this one. $F^{−1}(M)=\{(x,y,z):F(x,y,z)∈M\}=\{(x,y,z):f(x^2+y^2\le z)∈M\}=\{(x,y,z):x^2+y^2\le z∈f^{−1}(M)\}$
If anyone can help it would be very much appreciated!
Addition also would it be possible to prove $0<\lambda^3 (M)< \infty$ without explicitly calculating the measure? Can I use $\mu ((a,b])=b-a$ rule?
$M$ is a closed set. To see this, notice that $M = f^{-1}((-\infty, 0]) \cap g^{-1}((-\infty, 0])$, where $f, g : \mathbb R^3 \to \mathbb R$ are the functions defined by $f(x, y, z) = x^2 + y^2 - z$ and $g(x, y, z) = z - 1$. Since $f$ and $g$ are continuous and $(-\infty , 0]$ is closed, $f^{-1}((-\infty, 0]) \cap g^{-1}((-\infty, 0])$ is also closed.
All closed sets are Borel measurable.