REPRESENTING FINITELY ADDITIVE INVARIANT PROBABILITIES
STANFORD UNIV CA DEPT OF STATISTICS
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Hewitt and Savage have shown that finitely additive exchangeable probabilities on a product space are integral averages of power product probabilities. They prove this result as a corollary to their theorems on the countably additive case. This note adapts their technique to the study of more general invariant probabilities. From results of Farrell and Choquet and Feldman it is concluded that finitely additive invariant probabilities are averages of finitely additive ergodic probabilities. In a countably additive context it seems necessary to impose restrictions on the Borel field being studied and on the maps used to define invariance and ergodicity. Relaxing the assumptions of one type must be balanced by strengthening those of the other. Here, however, the field of sets can be arbitrary, and the maps are assumed only to be measurable. Rather than state a host of theorems which can be proved, one particular case is proved in detail. It is explained how the techniques can be applied to other problems. Several definitions of ergodicity are porposed and related to the one used. The final section contains a subjective probability interpretation of invariance and ergodicity.
- Statistics and Probability