In Gunning's book Introduction to Holomorphic Functions of Severals Variables, Vol I, on pages 44 and 45 have some statements about integrals that I can not understand.
First, on page 44, equation $(3)$ we have the integral
$\iint_{D_{r}} \frac{\partial f(\zeta)}{\partial \overline{\zeta}} \frac{d \overline{\zeta} \wedge d \zeta}{\zeta - z},$ where f is a $\mathcal{C}^{\infty}$ function in an open neighborhood of $\overline{D}$, $D$ an open subset of the complex plane bounded by a rectifiable simple curve $\gamma$, $z \in D$ and $D_{r} = D - \overline{\Delta}(z; r)$ for $r$ such that the disc is in $D$.
Gunning says that since $(\zeta - z)^{-1} d \overline{\zeta} \wedge d \zeta$ is a bounded measure in the plane, the integral above converges to the integral over $D$ as $r$ tends to zero. I can not understand $(\zeta - z)^{-1} d \overline{\zeta} \wedge d \zeta$ as a measure, much less bounded in the plane.
On page 45, we have the integral
$g(z) = \frac{1}{2\pi i} \iint_{\mathbb{C}^{1}} \frac{f(\zeta)}{\zeta - z} d \zeta \wedge d \overline{\zeta},$ where $f$ is a $\mathcal{C}^{\infty}$ function that vanishes outside a compact subset of $\mathbb{C}$.
Gunning says that this function is $\mathcal{C}^{\infty}$ in $\mathbb{C}^{1}$ (if $f$ is holomorphic, then $g$ is holomorphic). Why, if we have $(\zeta - z)$ in the denominator? I think this is related to the first question.
Perhaps I just need references to study more about integrals. Thanks.