The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" where he proves a (weak type) Bernstein Inequality using certain properties of `harmonic measure'.

I am not familiar with harmonic measure (I have checked wiki and got hold of a mammoth book by koosis (logarithmic integral) which has a section about harmonic measure). What I am looking for is the statements of theorems and principles about harmonic measures which are being used in the following argument.

Let $h(\xi)$ be the harmonic measure of the set $\mathbb R$ \ $[-y,y]$ with respect to the upper half-plane and a point $\xi \in \mathbb C_+$.

Let $z_1, z_2, \dots, z_{n_1}$ be such that $ Im (z_j) \leq 0$, and $\sum_1(z)=\sum_{j=1}^{n} \frac{1}{z-z_j}$.

Define $u(z) := h(-\sum_1(z))$.

The function $u(z)$ is harmonic in $C_+$, $0\leq u(z) \leq 1$, $u(it) \lim_{t\rightarrow + \infty} 0$, and $u(z) \geq 1/2$ if $|\sum_1(z)| \geq y$ (the latter fact follows from the geometric description of the harmonic measure as a ratio to $\pi$ of the angle at which a subset of $\mathbb R$ is seen from the point $\xi$).

Moreover, we have

$$ \lim_{t\rightarrow +\infty} \pi t u(it) =\int_{\mathbb R} u(x) dx \geq \frac{1}{2} \mu \{ x \in \mathbb R : |\sum_1(x)|>y \}.$$

On the other hand, an easy computation shows that

$\lim_{t\rightarrow \infty} \pi t u(it) =\lim_{t \rightarrow +\infty} \pi t h(\iota n/t + O (1/t^2)) = 2n/y.$

(As you can see by the end of this the author has obtained a weak type Bernstein Inequality).

I would also like to know if there are some lecture notes about harmonic measures available which is self contained and fits the category `every analysis student must know'.

Thankyou for your time and patience.