In probability theory one is sometimes tempted to ask for a uniform probability distribution over the reals. There isn’t one. The integers are no better here. I propose here a notion of hyper distribution, which is not a probability distribution, where there is a uniform (hyper) distribution over the reals, but not the integers. We will work up to a precise definition of hyper distribution by putting requirements on it, and then showing that such mathematical entities exist. We attempt to preserve a modified concept of pdf but that might fail to survive. In this text “uniform distribution” is not to be taken as “probability distribution”. We reserve “probability distribution” for when the pdf integrates to 1. This is all to find a better resolution to Bertrand’s ‘Paradox’. (The only thing paradoxical I see to that is the notion that every well specified set comes naturally with a uniform probability distribution.)

The first thing about a probability distribution that we must forego is choosing a sample from that distribution. We will retain weaker operations, however.

Warning I took a beautiful course in probability theory based on Lebesgue measure. I will try to explain anything that comes from that direction. That will tell some where I am going. (Clue: A Borel set is measurable, but what is important is that the intersection of a Borel set with any interval is measurable.)

This note describes some ways to choose points in a space too big to fit the classic definitions of a probability distribution where the measure of some total space is 1. Perhaps the section on Brownian functions should come here too.

Suppose we want to describe a set of points on the infinite plane ℝ2. We want the points to have some density ρ. We want the density to be asymptotically uniform. To get this we divide the plane into unit squares and deliver results incrementally. When asked for points from our sample from some bounded region, we consider the smallest set of squares that cover that region and if there are any new squares we have not considered, we choose a count of how many points to put in that square from the Poisson distribution. We then choose that many points uniformly within that square and can now give that response.

One of the methods that Bertrand gave for choosing a random chord had the pregnant property that if chords were chosen from a larger encompassing circle by that method, then those chords which intersected the small circle would supply chords to the small circle as it they had been chosen by that method. The circles need not be concentric. That suggests an infinite background of lines throughout the plane rather like a distribution with some sort of constant uniform PDF. But there are no such probability distributions. We posit a hyper distribution.
A rather more complicated example was considered by Levy as he created the notion of an n dimensional Brownian function. In nD he …