This is inspired by What results would follow from or imply "randomness" of the primes? , but I think it is sufficiently different to ask separately.
We can formalise probability in terms of an algebra $A$ of random variables equipped with an expectation function $E:A\to\mathbb{R}$ subject to certain axioms. For example, we can consider the algebra $A(n)$ of conjugation-invariant functions on $U(n)$, equipped with the Haar integral. We could then take some kind of limit $A(\infty)$. I don't know exactly what the details of the limiting process should be. Alternatively, we can take $B(a,b,c)$ to be some algebra of functions of arithmetical interest on the set $\{a,a+1,\dotsc,a+b\}^c$, where we think of $b$ as being quite large, and $a$ as being much larger. If we choose these algebras the right way, then it should be possible to compare $B(a,b,c)$ with $B(a',b',c')$ for suitable values of $a'$, $b'$ and $c'$. The things that people say about primes and random matrices suggest to me that it should be possible to construct some kind of limiting object $B(\infty)$, and to conjecture that $A(\infty)$ is isomorphic to $B(\infty)$ as algebras-with-expectation; and hopefully this would imply all the other conjectures that people talk about. Perhaps $B(\infty)$ could be described in terms of adeles.
Has anything like this been done? Or is it known to be impossible?
