How to apply Fubini's theorem in proof of Osgood's lemma

Question

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots z_n^{v_n}}{\zeta_1^{v_1+1}\ldots \zeta_n^{v_n+1}}d\zeta_1\ldots d\zeta_n,$$ (call the integrand $g$) for $z\in{\triangle (w,r)}.$ After this step i am supposed to change the summation and integration. But i am not able to justify this change.\ MY ATTEMPT: I am taking space $X=\mathbb{C}^n$ with product measure=$\mu$ and $Y=\mathbb{N}^n$ with counting measure=$\nu$. For Fubini's theorem i need to show that $g$ is integrable i product measure $\mu \times \nu.$ Can anyone help me in proving this. Or is there any other method to see this.