I like your idea. I think you are trying to transfer the uniform distribution on [0,1) over to Z. I am still trying to understand it. On Sun, 10 May 2020 at 19:29, Dan Asimov <dasimov@earthlink.net> wrote:
The concept of picking just one point at random from a probability distribution is not a rigorous mathematical concept, although maybe it could be made rigorous. Or maybe for some reason it cannot be.
Could you please explain further what you mean?
One bit of mathematical folklore is that there's no such thing as a random integer, if all integers are treated symmetrically.
i.e. a uniform probability distribution on the integers does not exist.
The cosets
C/G = {x + G | x in C}
of G form an uncontable quotient group that is rarely if ever discussed among decent mathematicians.
Let A denote a set consisting of exactly one element from each coset x + G.
Does the existence of A require the Axiom of Choice?
Then the subsets
A_n = A + g_n
of G form a partition of G by countably many sets that are rotated versions of each other.
subsets of C, did you mean?
Finally, since it is generally accepted that one can select a point at random from the uniform distribution on the unit interval [0, 1), which we identify with C, suppose this is done and the point selected is x.
i.e. The uniform probability distribution on [0,1) does exist (and the probability of each singleton subset {x} is 0). What would be the image of a subinterval, say [0,1/2], under phi? I can't see it. It should be half of Z, shouldn't it? I think it depends on the choice of A. Gary McGuire