Canonical Representation of Set Functions

The representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between no-necessarily additive set functions on the players space and additive ones on the coalitions space. (Or, alternatively, between nonadditive probability measures on a state space and additive ones on the space of events.) We extend the unanimity-basis representation to general (infinite) spaces of players, study spaces of games which satisfy certain properties and provide some conditions for α-additivity of the resulting additive set function (on the space at coalitions). These results also allow us to extend some representations of the Choquet integral from finite to infinite spaces.