A Note on Fine and Tight Qualitative Probabilities

Abstract

Savage (1954) has shown that fine and tight qualitative probabilities are realizable by finitely additive probability measures. His proof for this result is, however, in need of a correction. Fine qualitative probabilities are either atomless or equivalent to the union of $n$ equivalent atoms. Tight qualitative probabilities are always atomless. Qualitative probability structures, which are equivalent to the union of $n$ equivalent atoms, are realizable by a unique probability measure. Fine qualitative probabilities are almost realizable. With these results, the proof for Savage's theorem can be worked out and a theorem of Villegas (1964) can be strengthened.