A foundation for probabilistic beliefs with or without atoms

Abstract

We propose two novel axioms for qualitative probability spaces: (i) unlikely atoms , which requires that there is an event containing no atoms that is at least as likely as its complement; and (ii) third‐order atom‐swarming , which requires that for each atom, there is a countable pairwise‐disjoint collection of less‐likely events that can be partitioned into three groups, each with union at least as likely as the given atom. We prove that under monotone continuity , each of these axioms is sufficient to guarantee a unique countably‐additive probability measure representation, generalizing work by Villegas to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.