Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$. Suppose we have pairwise disjoint closed disks $D_n \subset \mathbb{D}$ of radius $r_n$ for $n \in \mathbb{N}$ such that $\sum_{n=1}^{\infty}r_n < \infty$. Show that then $A = \mathbb{D} \setminus \bigcup_{n=1}^{\infty}D_n$ has positive Lebesgue measure (we identify $\mathbb{C}$ with $\mathbb{R}^2$).
For $x \in (-1,1)$, let $n(x)$ denote the number of $D_n$ (possibly infinite) such that $z \in D_n$ for some $z$ with $\textrm{Re}(z) = x$. let $E = \{x \in (-1,1) : n(x) = \infty\}$. If we let $E_n = \{\textrm{Re}(z) : z \in D_n\}$, then $$E = \bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}E_n.$$ We have $\lambda(E_n) = 2r_n$ thus $\lambda(E) \leq \sum_{n=k}^{\infty}2r_n$ for each $k \in \mathbb{N}$. Since the $r_n$ are summable, this implies that $\lambda(E) = 0$ (here $\lambda$ denotes Lebesgue measure).
If $n(x) < \infty$, then since the disks are compact and disjoint, there is a minimum distance $\delta$ between any two distinct disks which contain points $z$ with $\textrm{Re}(z) = x$. By taking a point $z$ on the boundary of one the disks with $\text{Re}(z) = x$ and considering the vertical line segment joining $z$ and $z + i\delta$ or $z - i\delta$, we see that the set $\{\textrm{Im}(z) : z \not\in \bigcup_{n=1}^{\infty}D_n, \textrm{Re}(z) = x\}$ has positive Lebesgue measure.
This shows that for fixed $x \in (-1,1) \setminus E$, the characteristic function $\chi_A$ as a function of $y$ takes the value $1$ on a positive measure set. Since $(-1,1) \setminus E$ has positive measure, integrating $\chi_A$ with Fubini shows that $A$ has positive measure.