 However, a probability function that satisfies finite additivity without satisfying countable additivity is mathematically much more complicated than one that satisfies countable additivity. To even prove the existence of such a function over the algebra of subsets of a countable set of states requires the Axiom of Choice. With countable additivity, it is possible to specify a discrete probability function by enumerating the probabilities of the countably many states, and it is possible to specify a continuous probability function by enumerating the probabilities of the countably many rational open sets. But if merely finite additivity is assumed, specifying a probability function even on a countable state space may require specifying the probabilities of uncountably many events, rather than calculating the probabilities of these events from the countably many probabilities of the states. Furthermore, with such probability functions, many standard convergence results like the Strong Law of Large Numbers fail.