The question addressed in this paper is what kinds of welfare criteria
are sustainable, when the future states of the world evolve according to
a stochastic process. A stochastic process determines an infinite
sequence of ex-ante probability distributions over the states of the
world. It is shown that when a social welfare order over such sequences
is complete, transitive, continuous, and gives no dictatorship either to
the present or the future, then it is represented by a convex
combination of an integral over a countably additive measure and an
integral over a purely finitely additive measure. The notions of
symmetric treatment of the present and the future, stationarity for the
present and anonymity for the future are introduced. According to the
symmetric treatment, the distributions of the states of the world in the
present, when constant in time, and in the distant future can be
interchanged without affecting the welfare. The sustainable social
welfare order that treats the present and the future symmetrically and
satisfies stationarity for the present and anonymity for the future is a
sum of the discounted average of the expected utility and the expected
utility over the occupation measure of the stochastic process.