Let $f(x,y)$ be a positive, differentiable function defined in the unit disc $D\subset \mathbb{R}^2$ and let
$S=\{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2<1, z=f(x,y)\}$
Given $A \subset S$, Borel subset of $S$, define $\pi (A):=\{(x,y)\in D \mid (x,y,f(x,y)) \in A\} \subset D$, and
$$m(A)=\int_{\pi(A)} \sqrt{f_x^2+f_y^2+1} \, dx \, dy$$
Show that $m(\cdot)$ is Borel regular measure in $S$.
To show it is Borel measure, we have to show that every Borel set $A\subset S$ is $m(\cdot)$ measurable. To show it is $\it{regular}$, we have to show that for every set $A ⊆ S$ there exists a $m(\cdot)$ measurable set $E$ such that $A ⊆ E$ and $m(A) = m(E)$. If $A ⊆ E$, then, I have to show that
$$m(E)-m(A)=\int_{\pi(E) -\pi(A)} \sqrt{f_x^2+f_y^2+1} \, dx \, dy=0$$
Since $\sqrt{f_x^2+f_y^2+1}\subset \mathbb{L}_1(S,\mu)$, where $\mu(\cdot)$ is Lebesgue measure, the above equation means absolutely continuity which implies
$$\mu (\pi (E)-\pi (A))=0$$
Then, what does the above equation imply on? How I should continue?
Please leave me a comment if you know how I should proceed.