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 ℝ^{2}.
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 …