Additive representation of separable preferences over infinite products

Let X be a set of states, and let I be an infinite indexing set. Our first main result states that any separable, permutation-invariant preference order (>) on X^I admits an additive representation. That is: there exists a linearly ordered abelian group A and a `utility function' u:X-->A such that, for any x,y in X^I which differ in only finitely many coordinates, we have x>y if and only if the sum of [u(x_i)-u(y_i)] over all i in I is positive. Our second result states: If (>) also satisfies a weak continuity condition, then, for any x,y in X^I, we have x>y if and only if the `hypersum' of [u(x_i)-u(y_i)] over all i in I is positive. The `hypersum' is an infinite summation operator defined using methods from nonstandard analysis. Like an integration operator or series summation operator, it allows us to define the sum of an infinite set of values. However, unlike these operations, the hypersum does not depend on some form of convergence (recall: A has no topology) ---it is always well-defined. Also, unlike an integral, the hypersum does not depend upon a sigma-algebra or measure on the indexing set I. The hypersum takes values in a linearly ordered abelian group A*, which is an ultrapower extension of A. These results are applicable to infinite-horizon intertemporal choice, choice under uncertainty, and variable-population social choice.