Let $v$ be a complex Borel measure on $[0,1]$ and $m$ be the Lebesgue measure. We define the support of measure by $$supp(v) = [0,1]-\cup\{I \subset [0,1]: v(I)=0\}$$ where $I$ is an interval.
Suppose that $m(\{supp(v)\})=0$. Show that for a.e. $\theta \in [0,1]$ the following limit exists. $$\lim_{\epsilon \to 0}\int_{|\theta -\phi|>\epsilon}\cot{(\pi(\theta-\phi))}dv(\phi)$$
The cotangent has a good cancellation property but it seems to be hard to use it because $v$ is not translation-invariant. I can't progress any more. I need your help.