Runge's theorem

Let $\Omega = \left\{ z:\left|z\right| < 1\text{ and }\left|2 z-1\right| > 1\right\}$, and suppose $f \in H(\Omega )$.

Is there a sequence of polynomials $P_{n}$ such that $P_{n}(0) = 1$ for $n = 1,2,3, \ldots$, but $P_{n}(z) \rightarrow 0$ for every $z \neq 0$, as $n \rightarrow \infty$?

Is there a sequence of polynomials $P_{n}$ such that

For $n = 1,2,3, \ldots$, let $\Delta_{n}$ be a closed disc in $U$, and let $L_{n}$ be an arc (a homeomorphic image of $[0,1])$ in $U-\Delta_{n}$ which intersects every radius of $U$. There are polynomials $P_{n}$ which are very small on $\Delta_{n}$ and more or less arbitrary on $L_{n}$. Show that $\left\{ \Delta_{n}\right\} ,\left\{ L_{n}\right\}$, and $\left\{ P_{n}\right\}$ can be so chosen that the series $f = \Sigma P_{n}$ defines a function $f \in H(U)$ which has no radial limit at any point of $T$. In other words, for no real $\theta$ does $\lim_{r \rightarrow 1} f\left(r e^{i \theta }\right)$ exist.

Show that in Theorem 13.9 we need not assume that $A$ intersects each component of $S^{2}-\Omega$. It is enough to assume that the closure of $A$ intersects each component of $S^{2}-\Omega$.

Suppose, however, that $f$ is any complex-valued measurable function defined in the complex plane, and prove that there is a sequence of holomorphic polynomials $P_{n}$ such that $\lim_{n \rightarrow \infty } P_{n}(z) = f(z)$ for almost every $z$ (with respect to two-dimensional Lebesgue measure).

Extend Mergelyan's theorem to the case in which $S^{2}-K$ has finitely many components: Prove that then every $f \in C(K)$ which is holomorphic in the interior of $K$ can be uniformly approximated on $K$ by rational functions.