Open set having boundary with non-zero hyperbolic area

Question

I encountered the following statement in Katok's Fuchsian Groups:

Theorem 3.1.1: Let $F_1$ and $F_2$ be two fundamental regions for a Fuchsian group $\Gamma$, and $\mu(F_1) < \infty$. Suppose that the boundaries of $F_1$ and $F_2$ have zero hyperbolic area. Then $\mu(F_1) = \mu(F_2)$.

A closed region is defined in the preceding definition as being the closure of a non-empty open set, and a fundamental region is just a closed region satisfying certain properties. Further, $\mu(A)$ is the hyperbolic area of $A$, defined as $$\mu(A) = \int_A \frac{\mathrm{d}x \mathrm{d}y}{y^2}.$$

My question is: can there exist a fundamental region (or even just a closed region, or equivalently an open set) having a boundary with non-zero hyperbolic area (and if yes, is there an example I could look at)? Or was this hypothesis included in the statement of the theorem to avoid proving that this cannot happen?

Additionally, are the notions of 'measure' and 'area' the same here?

Edit: Katok defines a fundamental regions as follows: Let $X$ be a metric space, and $G$ be a group of homeomorphisms acting properly discontinuously on $X$. Then $F \subset X$ is a fundamental region for $G$ if:

1. $\bigcup_{T \in G} T(F) = X$,
2. $\mathring{F} \bigcap T(\mathring{F}) = \emptyset$ for all $T \in G \ \{\mathrm{Id}\}$.

Here, $\mathring{F}$ denotes the interior of $F$.

Answer 1

There are simple curves in the Euclidean plane which have positive 2-dimensional Lebesgue measure. (Jordan curves with this property are called "Osgood curves.") Similarly, given two distinct points $x_1, x_2$ in the real axis, there is a simple arc $A$ of positive 2-dimensional Lebesgue measure connecting $x_1, x_2$ contained (apart from its end-points) in the upper half plane. Here by the upper half-plane I mean $$ U=\{z\in {\mathbb C}: Im(z)>0\}. $$ (I will be using $U$ as a model of the hyperbolic plane in the usual manner.) Accordingly, $A$ will have positive hyperbolic area (hyperbolic measure and Lebesgue measure are absolutely continuous with respect to each other). Without loss of generality, $x_1=-1, x_2=1$. There exists $\lambda> 1$ such that the hyperbolic isometry $T_\lambda: z\mapsto \lambda z$ satisfies: $$ A\cap T_\lambda A=\emptyset. $$ Thus, the arcs $A, T_\lambda A$ bound a region $D\subset U$ which is a fundamental domian for the action of the discrete group $\Gamma=\langle T_\lambda \rangle$ on $U$. This domain has the property that its boundary has positive 2-dimensional Lebesgue measure (equivalently, positive hyperbolic area).