i have to show that the complex log is mesurable (borel): $\displaystyle log: \mathbb{C} \ {0} \rightarrow \mathbb{C} z=re^{i\phi}\rightarrow ln r+i\phi$

A collague told me to start this way: $\displaystyle log^{-1}(B)=log^{-1}(B \cap \bigcup_{n\in \mathbb{N}} [-n,n]\times[0,2\pi-\frac{1}{n}] )$.

How do i know that? The rest i think is easy but my problem is with this union! Is the complex log only on $\displaystyle [-n,n]\times[0,2\pi-\frac{1}{n}] $ defined?

Thank you in advance