Consider a (homogeneous) spatial Poisson process $\Pi$ in $[0,1]^2$ with constant rate $\lambda^2$. For each $x \in \Pi$, let $D_x$ be the disc centered at $x$ with radius $r$, and let $D = \bigcup_{x \in \Pi} D_x$ be the union of all disks.
We couple $r$ and $\lambda$ such that $\pi r^2 \lambda^2 = \varepsilon$ for some $\varepsilon \in (0,1)$ fixed, so that the (approximate) expected area of $D \cap [0,1]^2$ is constant ($\pi r^2$ is the area of one disk and $\lambda^2$ is the expected number of points in $\Pi$).
I am interested in the $r \rightarrow 0$ (equivalently, $\lambda \rightarrow \infty$) limit of this process. In particular, I suspect that the indicator function $1_{D \cap [0,1]^2}$ of the region $D$ in $[0,1]^2$ tends to the constant function $\varepsilon$ in the limit, in the weak sense. That is, in the limit the collection of disks $D$ end up 'blanketing' the space $[0,1]^2$ with density $\varepsilon$.
I am unsure if this is an established property in the literature and do not know how to go about proving this. We can upper-bound $\text{Area}(D)$ by $|\Pi|\cdot \pi r^2$, which tends to $\varepsilon$ in the limit, but I cannot get anything more.