The existence of security market equilibrium with a non-atomic state space

We consider a finite horizon economy with incomplete markets and, at each period, a non-atomic continuum of states of nature, We combine backward recursion methods with a reformulation of the problem in terms of first-order conditions to establish the existence of a Radner equilibrium. One of our hypotheses, not required in the case of a finite number of states, says that in any date and state the position reached after any possible previous asset trades, but before spot trades, constitutes a feasible consumption. An example shows that this hypothesis cannot be dispensed with (even in the case of countable many states).