I wonder if there exist ways to deal with infinitely coutable sets with a probability in the "uniform" sense. In the basic Probability Theory, as far as I know, we usually study two kinds of probability most of the time that is compatible with our sense of probability:
1- First one is the discrete probability which uses convergent sums and mass functions defined on a finite outcome space $\Omega$ and finite $\sigma$-algebra $\mathcal{F} \subset 2^{\Omega}$ such that $\mathbb{P}(\Omega) = 1$. We consider $\sigma$-algebras to generalize the concept but things are simpler here and all can be induced to positive sums and series.
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In this version, uniform distributions corresponds to the case that the mass function is constant and $f({x}) = 1/n$ for each outcome $x\in\Omega$ where $|\Omega| = n$ where $\mathcal{F}$ is just the entire power set $2^{\Omega}$ since it should contain all singletons and all sets are finite. Then, for any event $E\in \mathcal{F} = 2^{\Omega}$ where $|F| = k \leq n$, it produces $f(E) = k/n$ by finite additivity of the (probability) measure.
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Other than the uniform distribution, we can consider any countable $\Omega$ where $\mathbb{P}(\Omega) = 1$ is still satisfied which means that any non-negative function $f$ satisfying $\sum_{x\in \Omega} \mathbb{P}({x}) = 1$ can be considered as a mass function. Then, the sense for uniform distribution does not work here as a probability mass function because countable infinite set causes the mass function $f(x)$ to be zero if $E$ is finite or undefined caused by indeterminate quotient $\infty/\infty$ if $E$ is infinite.
2- However, there is the second kind of the theory, the continuous probability on, for example, real numbers $\mathbb{R}$ using Lebesgue Integration. In this case, probability is not considered as the sum of singleton outcomes, i.e. $\sum f(x)$, but the integral of the density function of an event $E\in \mathcal{F}\subset 2^{\Omega}$ as the integral $\int_E fd\lambda$ in general. So we can use the similar set-up here where the only change is to use integration instead of sums and series.
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Now the sense for uniform density here is just a constant function again: Given a bounded measurable set E which is not null, e. g. intervals. Then $c := m(E) \in (0, \infty)$ and we can use the uniform distribution on this bounded set as $f(x) = 1/c > 0$. Then we simply have the total probability as the integral $\int_E f dm = (1/c)c = 1$ which makes it a distribution for real.
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Unfortunately, there are things we are missing here, too: None of the null sets are covered by the continous probability as well as sets with measure $\infty$:
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The latter is similar to the problem we encountered in discrete case since it causes the uniform distribution on the set to be 0 and then the probibility, which means the integral of the zero distribution here, becomes 0, not 1. On the other hand the former is caused by the fact that integrals over null sets are zero automatically, not 1. Thus no function $f$ can be a density function for a null set like countable infinite sets $\mathbb{N}$ or $\mathbb{Q}$ if they are null in the given measure, for example Lebesgue Measure or Borel measure.
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Finally, I think that using a measure where the set $\mathbb{N}$ is
not null (consider a measure where every natural number is an
atom that of equal measure) coincides with the discrete probability with some editing since the integration
on $\mathbb{N}$ becomes the limit of the finite sum by the definition
of the integral via simple functions.
QUESTION: We have the sense that choosing a random natural number should be considered in some sort of "uniform" way. How can we describe that uniformity and study it in a natural sense? Which tools do we have to examine them?
After my surfing on that topic I realized that the discrete way cannot handle some of infinite quantities includig uniform ones and continous way cannot handle null-sets including countable ones. So I believe that there should be a middle-way or a completely different approach to study (infinitely) countable uniformity.
Best Answer
You might find a paper by Bumby and Ellentuck on 'Finitely-additive Measures and the First Digit Problem' worth looking at. Bumby has slides here giving a some-years-later review of that paper and some of the followup. This is at least one approach to the problem you're thinking about.
As a small side-remark, I know about this paper because when I was an undergraduate in the 1970s, I wondered the same thing, and my professor suggested I look at this paper --- so this is apparently a question that keeps arising over the years. As I recall, I actually called up and spoke to one of the authors, who said that the original manuscript of the paper used non-standard analysis, but the reviewers rejected it on that basis, so they had to unwind the definitions to write the eventually-accepted version. That explains the presence of all that stuff about ultrafilters, etc.