Let $\mathrm{Rect}$ denote the class of axis-parallel rectangles $r: \mathbb{R}^2 \to \{0,1\}$, assigning $1$ if the point is inside the rectangle and $0$ otherwise. Let $\mathcal{D}$ be a distribution defined on the Borels of $\mathbb{R}^2$. $(\mathrm{Rect}, d_{L^1_\mathcal{D}})$ is a pseudo-metric space, where $d_{L^1_\mathcal{D}}(r,r') = \mathbb{P}_{\mathcal{D}}[r \Delta r']$ which is the probability of belonging to rectangle $r$ and not $r'$ (and vice versa).
Question 1: Show that for all $\epsilon > 0$, the rectangles whose diagonals points are given by the $$\Biggl\{(x_k,y_k'): \mathbb{P}\Big\{]-\infty,x_k] \times \mathbb{R} \Big\} = \frac{k \epsilon}{8}, \mathbb{P}\Big\{\mathbb{R} \times ]-\infty,y_{k'}] \Big\} = \frac{k' \epsilon}{8} \Biggl\}_{(k,k') \in \{0, \dots, \lfloor \frac{8}{\epsilon} \rfloor \} }$$ form an $\epsilon$-net of $\mathrm{Rect}$.
Question 2: what is the general approach to constructing $\epsilon$-nets over the class $\mathrm{Rect}$? It seems like the idea is to start from an $\epsilon$-net over $[0,1]$ to cover the possible probability values since we use the pseudo-metric defined by the distribution $\mathcal{D}$.
Question 1
For question 1 the idea is to reduce the infinite space of $\mathbb{R}^2$ so that any rectangle outside the plane $<x_0,y_0,x_{k^*}, y_{k^*}>$ has 0 probability of occurring, where $k^* = \lfloor \frac{8}{\epsilon} \rfloor$, I tried to use the double integral to bound the probability, however, this will require 4 variables $k_1,k_2,k_3, k_4$. Is there an easy fast way to show the result?
