Injectivity of the Dubins-Freedman construction of random distributions Summary: Injectivity of the Dubins-Freedman 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 upper-right rectangle and iterate the process there; and do the same for the lower-left 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 one-to-one. (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 Collections: Mathematics