We define a natural, computable map that associates to each real a Borel probability measure, so that we can talk about random measures, the images of the (ML) random reals. We show that such random measures are atomless and mutually singular with respect to Lebesgue. We introduce a certain property that lies strictly between atomlessness and absolute continuity (with respect to the Lebesgue measure) that we conjecture random measures satisfy. We then discuss other maps, ask the question "Why is the first map more natural?", and discuss some ways in which that question might be precisifiable.