A foundation for probabilistic beliefs with or without atoms

We provide sufficient conditions for a qualitative probability (Bernstein, 1917; de Finetti, 1937; Koopman, 1940; Savage, 1954) that satisfies monotone continuity (Villegas, 1964; Arrow, 1970) to have a unique countably additive measure representation, generalizing Villegas (1964) to allow atoms. Unlike previous contributions, we do so without a cancellation or solvability axiom. First, we establish that when atoms contain singleton cores, unlikely cores—the requirement that the union of all cores is not more likely than its complement—is sufficient (Theorem 3). Second, we establish that strict third-order atom-swarming—the requirement that for each atom A, the less likely non-null events are (in an ordinal sense) more than three times as likely as A—is also sufficient (Theorem 5). This latter result applies to intertemporal preferences over streams of indivisible objects.