$M$ is a surface of class $C^1$.
$L^2(M)$ is the set of 1-forms in M with the following properties:
1) $\omega$ is measurable locally i.e for each chart $(U_\alpha,z_\alpha)$, $\omega$ is $u_\alpha dz+v_\alpha \overline{dz}$ with $u_\alpha,v_\alpha$ measurables
2) there exists a open $A\subseteq M$ such that $A$ is second countable and $\omega\mid_{M\setminus A}=0$ almost everywhere i.e for each chart $(U_\alpha,z_\alpha)$, $u_\alpha\mid_{z_\alpha(U_\alpha\setminus A)}=0$ and $v_\alpha\mid_{z_\alpha(U_\alpha\setminus A)}=0$ almost everywhere.
3) $\displaystyle \int_A \vert \omega(z)\vert ^2 \vert dz\vert ^2<\infty$
I has two questions:
1) what does it mean that $u_\alpha,v_\alpha$ are measurables? $u_\alpha,v_\alpha$ are defined in $A\subseteq M$
2) what does it mean the integral in 3) the symbol $\vert dz\vert ^2$ and the definition of the integral ?
Thanks you all