Let $\mathbb{H}$ be the open upper half of the complex plane, that is, $$\mathbb{H} = \{ z \in \mathbb{C} : \text{Im}(z) > 0\}.$$ For $x > 0$, define $$\Omega_x = (\mathbb{H}+i/2) \cup \{z \in \mathbb{H} : \text{Im}(z) \leq 1, \text{Re}(z) < x\}$$ Also, define $$F_x = \{x+ti : t \in [0,1/2]\} \cup \{t+i/2 : t \in [x,+\infty)\} \subset \partial\Omega_x.$$ Maybe a picture helps:
My interest lies in the function
$$x > 0 \mapsto \omega(i,F_x,\Omega_x).$$
This function is clearly decreasing in $x$, and so
$$\exists L = \lim_{x \to + \infty}\omega(i,F_x,\Omega_x) \in [0,1).$$
My question is: is there any argument to decided if $L = 0$ or $L > 0$?
My intuition says $L = 0$, but that would have some implications in the field I'm working on (discrete iteration of holomorphic maps) that seem not very plausible.
Edit: Let $\Omega \subset \mathbb{C}$ be a connected set, and $F \in \partial \Omega$ be measurable. Then, for any $z \in \Omega$, we define $$w(z,F,\Omega)$$ as the harmonic measure of $F$ respect to $z$, that is, the value at point $z$, $u(z)$, of the harmonic function $u$ that solves the problem $$\text{(P) } u \text{ harmonic on } \Omega, \quad u = \chi_F \text{ on } \partial \Omega,$$ where $\chi_F$ es the characteristic function of $F$.