I have been trying to understand the following fact, consider $\mathbb{H}$ the upper half plane of the complex numbers. And let $\Gamma_0 = SL_2(\mathbb{Z})$ act on $\mathbb{H}$ we know that there is a $\Gamma_0$ invariant measure on $\mathbb{H}$ defined as $\mu = \frac{dxdy}{y^2}$. A book by Anton Deitmar sates that this measure is invariant so we may define a measure on $\Gamma_0 \backslash \mathbb{H}$ which is the space of orbits, this fact is unclear to me could you help me understand this please?
The first possible definition I think of is to choose a fundamental domain $D$ for the action of $\Gamma_0$ on $\mathbb{H}$ and define the measure $\overline{\mu}(A) = \mu(q^{-1}(A) \cap D)$ where $q$ is the quotient map associated to $\Gamma_0 \backslash \mathbb{H}$. But the problem is that i do not know how to show that this definition does not depend on the fundamental domain $D$. Indeed if we have two fundamental domains $D$ and $D'$ we may not have $D' = gD$ with $g \in \Gamma_0$ if we choose a "weird" $D'$ (if we always have $D' = gD$ then it is trivial because we may use the invariance of $\mu$).