A foundation for probabilistic beliefs with or without atoms

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.