Then why is the notion of generalized Bayes rule an improvement? It is indeed equivalent to the notion of Bayes rule when a Bayes rule exists and all formula_39 have positive probability. However, no Bayes rule exists if the Bayes risk is infinite (for all formula_19). In this case it is still useful to define a generalized Bayes rule formula_19, which at least chooses a minimum-expected-loss action formula_64 for those formula_39 for which a finite-expected-loss action does exist. In addition, a generalized Bayes rule may be desirable because it must choose a minimum-expected-loss action formula_52 for "every" formula_39, whereas a Bayes rule would be allowed to deviate from this policy on a set formula_68 of measure 0 without affecting the Bayes risk.