Countably Additive Subjective Probabilities

The subjective probabilities implied by L. J. Savage's (1954, 1972) postulates are finitely but not countably additive. The failure of countable additivity leads to two known classes of dominance paradoxes, money pumps and indifference between an act and one that pointwise dominates it. There is a common resolution to these classes of paradoxes and to any others that might arise from failures of countable additivity. It consists of reinterpreting finitely additive probabilities as the 'traces' of countably additive probabilities on larger state spaces. The new and larger state spaces preserve the essential decision-theoretic structures of the original spaces.