The study of complete markets is central to state-preference theory.
Arrow and Debreu were awarded the Nobel Memorial Prize in Economics (Arrow in 1972, Debreu in 1983), largely for their work in developing the theory of complete markets and applying it to the problem of general equilibrium.
A state-contingent claim, or state claim, is a contract whose future payoffs depend on future states of the world.
For example, suppose you can bet on the outcome of a coin toss.
A bet on heads is a state claim, with payoff of one dollar if heads is the outcome, and payoff of negative one dollar if tails is the outcome.