Zero area of the Mandelbrot set boundary

The Mandelbrot set is the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $z \mapsto z^2 + c$ remains bounded; equivalently, it is the set $\{c \in \mathbb{C} : \forall k \in \mathbb{N}, \|z_k\| \leq 2\}$ where $z_0 = 0$ and $z_{k+1} = z_k^2 + c$. The frontier (topological boundary) of a set is the intersection of its closure with the closure of its complement. The volume (Lebesgue measure) of a measurable subset of $\mathbb{C} \cong \mathbb{R}^2$ is its two-dimensional Lebesgue measure. Prove that the volume of the frontier of the Mandelbrot set equals $0$.