The Mandelbrot set is the set of all complex parameters c for which the orbit of 0 under iteration of the map z↦z2+c remains bounded; equivalently, it is the set {c∈C:∀k∈N,∥zk∥≤2} where z0=0 and zk+1=zk2+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 C≅R2 is its two-dimensional Lebesgue measure. Prove that the volume of the frontier of the Mandelbrot set equals 0.