/sci/ - Science & Math

Let [math]\mu = \frac{dx dy}{y^2}[/math] be the standard measure on the upper half plane [math]\mathbb{H} = \{x + iy \mid y > 0\}[/math]. Fix an open set [math]U \subset \mathbb{H}[/math] which is contained in [math]A \times (\delta, \infty)[/math] for some bounded set [math]A \subset \mathbb{R}[/math] and [math]\delta > 0[/math], and thus has finite measure.

Given [math]\epsilon > 0[/math], can we always find a finite collection of pairwise disjoint hyperbolic triangles [math]T_1 \dots T_n[/math] such that [math]\mu(\bigcup_{i=1}^n T_i \triangle U) < \epsilon[/math]?