F. and M. Riesz theorem

Let $P = \left\{ p|\partial \mathbb {D} : p = \text{an analytic polynomial}\right\}$ 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 $P$.

A function in $\mathbf{H}^2$ vanishes either almost everywhere or almost nowhere.

Suppose $\mu$ is a complex Borel measure on $T$ such that

Prove that then either $\mu = 0$ or the support of $\mu$ is all of $T$.

Suppose $K$ is a proper compact subset of the unit circle $T$. Prove that every continuous function on $K$ can be uniformly approximated on $K$ by polynomials. Hint: Use Exercise 14.