Theory of Probabilistic Connectedness

We introduce and study a notion of probabilistic connectedness, which we term $proconnectedness$, defined in terms of partitions of a probability space into two nonempty disjoint independent events. Both proconnectedness and disproconnectedness are shown to be invariants (in a suitable sense) under isomorphic random elements. We show that a substantial part of the fundamental theory of topological connectedness admits a natural counterpart in the present theory of proconnectedness. Some applications and connections regarding limit theorems, cardinality equality of measurability structures, atomic distributions, and singular distributions are discussed.