Brownian local minima and other random dense countable sets

We compare two examples of random dense countable sets, Brownian local minima and unordered uniform infinite sample. They appear to be identically distributed. A framework for such notions is proposed. In addition, random elements of other singular spaces (especially, reals modulo rationals) are considered. Introduction For almost every Brownian path ω = (bt)t∈[0,1] on [0, 1], the set (0.1) Mω = {s ∈ (0, 1) : ∃ε > 0 ∀t ∈ (s− ε, s) ∪ (s, s+ ε) bs < bt} of local minimizers on (0, 1) is a dense countable subset of (0, 1). Should we say that (Mω)ω is a random countable dense set? Can we give an example of an event of the form {ω : Mω ∈ A} possessing a probability different from 0 and 1 ? No, we cannot (see also Corollary 5.1). All dense countable subsets of (0, 1) are a set DCS(0, 1) (of sets), just a set, not a Polish space, not even a standard Borel space. What should we mean by an DCS(0, 1)-valued random variable and its distribution? Apart from such conceptual questions we have specific examples and questions; here is one. A ‘uniform infinite sample’, that is, an infinite sample from the uniform distribution on (0, 1) may be described by the product ( (0, 1),mes ) of an infinite sequence of copies of the probability space (