Hahn-Banach theorem in boundary set of complex disc This is an exercise from Conway's book

*

*Let $P=\{p \mid \partial \mathbb{D}: p= \text{an analytic polynomial}\}$ and consider $P$ as a manifold in $C(\partial \mathbb{D})$. Show that if $\mu$ is a real-valued measure on $\partial \mathbb{D}$ such that $\int p d \mu=0$ for every $p$ in $P$, then $\mu=0$.

*Give an example of a complex-valued measure $\mu$ such that $\mu \neq 0$ but $\int p d \mu=0$ for every $p$ in $P$.

For part 1, I know that if $P$ in dense in $C(\partial\mathbb D)$ then $\int p d\mu = 0$ for all $p \in P$. However, I cannot find way to apply Stone-Weirerstass since I cannot show $P$ is self-adjoint. Another way that I can think is I can suppose
$$
p = x + i y
$$
where $x, y:\partial \mathbb D \to \mathbb R$. How do I argue that $x$ and $y$ are dense in all real-valued functions defined on $\partial \mathbb D$?
For part 2, I think I can apply the residue theorem. How do I define measure $\mu$?
 A: The set of polynomials is dense in $$L^{1}(\partial \mathbb{D},|\mu|),$$where $|\mu|$ is the variation of $\mu$ produced by the Hahn-Jordan decomposition theorem. As such, we may consider a $\mu$-measurable subset $U$ of the circle $\partial \mathbb{D}$ and a sequence of polynomials $\{p_{n}\}_{n}$ converging to the characteristic function $\chi_{U}$ of $U$ in the norm of $L^{1}(\partial \mathbb{D},|\mu|)$. Then $$ |\mu (U)| = \left| \int_{\partial \mathbb{D}} \chi_{U}d\mu \right| \leq \int_{\partial \mathbb{D}} |\chi_{U}-p_{n}|d|\mu| + \left| \int_{\partial \mathbb{D}} p_{n} d\mu\right|= \left\| \chi_{U}-p_{n} \right\|_{L^{1}(\partial \mathbb{D},|\mu|)}. $$ Letting $n\rightarrow \infty$ we obtain $\mu(U)=0$ for any measurable set $U$. This indicates that $\mu$ is identically zero.
There is a subtle point to be made here though. In order for $\chi_{U}$ to be in $L^{1}$ regularity should be assumed for the measure $|\mu|$.
As for the second question we may consider the "complexified" Lebesgue measure $$ dz=dx+idy, \qquad z=x+iy.$$ Since any polynomial $p$ is a holomorphic function and the circle fulfills the assumptions of Cauchy's theorem we get $$\int_{\partial \mathbb{D}}p(z)dz=0,$$ for any polynomial.
I hope this is helpful!
