The simplest version of expected utility formulates X as a set of finite-support probability distributions, also called lotteries , on a set C of consequences. The consequences in C are viewed as mutually exclusive outcomes of decision. It is commonly assumed that X is closed under convex combinations so , defined by for all , is in X whenever and . This is tantamount to assuming that X is closed under every binary operation in a set of operations for , with . A straightforward generalization of the simple version under convex closure takes X as a function set closed under a mixture operation which maps each triple for and into another member of X that is usually denoted by or or .
Independence says that is preserved under similar mixtures.
It is usually defended by an interpretation for and , whereby x or y obtains with probability and z obtains with probability . Continuity ensures the existence of real-valued utilities as opposed to nonstandard or multidimensional lexicographically ordered utilities. We discuss the lexicographic case in Section 5. Then, whenever and , we prove that for a unique . The representation is then extended to the rest of X in the only way that preserves linearity under indifference.