If the utility function is unbounded, any finite utility assignment to the St Petersburg game is consistent with expected utility theory

The von Neumann Morgenstern theorem gives necessary and sufficient conditions for a binary relation on a convex subset of probability measures to be represented by a linear functional on the vector space of signed measures. The axioms of expected utility theory thus require that all probability measures, including the St Petersburg game, are assigned utility values that are finite and respect linearity. Unbounded utility functions do not provide enough data to determine the representing functional. This requires specifying utility values on a Hamel basis of the vector space of signed measures, which may be non-constructive. Extending by continuity and truncation continuity can enlarge the domain where the expected utility representation is valid, but not to probability measures that have undefined expected utility. Utility values can thus be freely assigned to any linearly independent set of probability measures whose expected utility is not defined. In particular, any finite utility value assignment to the St Petersburg, Pasadena and related games is consistent with the axioms of expected utility theory.