As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful (and controversial) for ethical decisions. Utility function Consider a set of alternatives among which a person can make a preference ordering. The utility obtained from these alternatives is an unknown function of the utilities obtained from each alternative, not the sum of each alternative. A utility function is able to represent that ordering if it is possible to assign a real number to each alternative in such a manner that alternative a is assigned a number greater than alternative b if and only if the individual prefers alternative a to alternative b. In this situation someone who selects the most preferred alternative is necessarily also selecting the alternative that maximizes the associated utility function. Suppose James has utility function such that x is the number of apples and y is the number of chocolates. Alternative A has apples and chocolates; alternative B has apples and chocolates. Putting the values x, y into the utility function yields for alternative A and for B, so James prefers alternative B. In general economic terms, a utility function measures preferences concerning a set of goods and services. Utility is often correlated with concepts such as happiness, satisfaction, and welfare which are difficult to measure. Thus, economists utilize consumption baskets of preferences in order to measure these abstract, nonquantifiable ideas. Gérard Debreu precisely defined the conditions required for a preference ordering to be representable by a utility function. For a finite set of alternatives these require only that the preference ordering is complete (so the individual is able to determine which of any two alternatives is preferred or that they are equal), and that the preference order is transitive. Very often the set of alternatives is not finite, because