finding a complex- valued measure 1- Put $A= \Bbb D^{-}\cap {\Bbb D^{c}}^{-}$(Boundery of $\Bbb D$). Let $P= \{p|A; p=$ an analytic polynomial$\}$ and iconsider $P$ as a manifold in $C(A)$. Show that if $\mu$ is a real- valued measure on $A$ such that $\int pd\mu =0$ for every $p$ in $P$, then $\mu=0$.
2- Give an example of a complex - valued measure $\mu$ such that $\mu\neq 0$ but $\int pd\mu =0$ for every $p$ in $P$.
I know that $P$ is dense in $C(A)$, and also $M(A)\simeq C(A)^*$ so $\mu=0$. 
Please any hint.
Thanks so much.

Hint: Cauchy's integral theorem.
