Theoretical Economics 14 (2019), 709–778

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### A foundation for probabilistic beliefs with or without atoms

*Andrew Mackenzie*

#### Abstract

We propose two novel axioms for qualitative probability spaces (Bernstein, 1917; de Finetti, 1937; Koopman, 1940; Savage, 1954): (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 (Villegas, 1964; Arrow, 1970), each of these axioms is sufficient to guarantee a unique countably additive probability measure representation, generalizing Villegas (1964) to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.

Keywords: Beliefs, qualitative probability, unlikely atoms, atom-swarming

JEL classification: D81, D83

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