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could anybody will help me to do this problems:

  1. Let $\mathcal D$ be the unit disk a Set $E\subseteq\partial\mathcal D$ has harmonic measure identically $0$ with respect to $\mathcal D$. What can you conclude about $E$?
  2. Let $\Omega\subseteq\mathbb C$ be a domain and let $E \text{ and } E'\subseteq\Omega$ such that $\omega(z,\Omega,E)\ge\omega(z,\Omega,E') \forall z\in\Omega$. What can you conclude about $E$ and $E'$?

Problems are taken from Respected Stephen Krantz's Geometric function theory book chapter9.

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Hmm...this is incredibly open ended. I'm not sure what sorts of things it is looking for. If E has harmonic measure 0, then there is 0 probability that a Brownian motion started in D will first touch $\partial \mathcal{D}$ in the set $E$. – Matt May 22 '11 at 20:21

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