 
Summary: Injectivity of the DubinsFreedman construction
of random distributions
Pieter Allart
University of North Texas
In the 1960s, L. Dubins and D. Freedman introduced the following
simple method to generate a probability distribution on the unit inter
val at random. Starting with a base measure µ (a probability measure
on the unit square S = [0, 1]2
), pick a point at random according to
µ. This point divides S into four rectangles. Now map the measure
µ affinely onto the upperright rectangle and iterate the process there;
and do the same for the lowerleft rectangle. Iterating this procedure
ad infinitum one obtains the closed graph of a distribution function F.
This procedure defines a mapping µ Pµ from the space of base
measures to the space P of probability measures on , where is
the space of all distribution functions on [0, 1] (suitably metrized). An
unsolved question is to find the largest possible "natural" domain for
this mapping on which the mapping is onetoone. (It is clear that base
measures supported on the main diagonal of S must be excluded, as
must measures which give all their mass to two opposing points on the
