  The Minkowski Question Mark function relates the continued-fraction
representation of the real numbers, to their binary expansion. This function is
peculiar in many ways; one is that its derivative is 'singular'. One can show
by classical techniques that its derivative must vanish on all rationals. Since
the Question Mark itself is continuous, one concludes that the derivative must
be non-zero on the irrationals, and is thus a discontinuous-everywhere
function. This derivative is the subject of this essay.
  Various results are presented here: First, a simple but formal
measure-theoretic construction of the derivative is given, making it clear that
it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is
safe to manipulate in various familiar ways. Next, an exact result is given,
expressing the measure as an infinite product of piece-wise continuous
functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d).
This construction is then shown to be the Haar measure of a certain transfer
operator. A general proof is given that any transfer operator can be understood
to be nothing more nor less than a push-forward on a Banach space; such
push-forwards induce an invariant measure, the Haar measure, of which the
Minkowski measure can serve as a prototypical example. Some minor notes
pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made.
