INVARIANT EXTENSIONS OF LINEAR FUNCTIONALS WITH APPLICATIONS TO MEASURES AND STOCHASTIC PROCESSES

A theorem is proved slightly stronger than the following. Let G be a set of order-preserving linear operators on a partially- ordered real linear space X, for which there exist sets G = Gn ⊇ Gn−1 ⊇ … ⊇ G0 with G0 commutative and such that for k = 1,…, n, x in X, g1 and g2 in Gk there exist h1 and h2 in Gk−1 satisfying h1g1g2(x) = h2g2g1(x). If S is a G-invariant subspace such that for all x in X there is an s in S satisfying s ≧ x, and f0 is a G-invariant positive linear functional on S, then f0 extends to a G-invariant positive linear functional on X. This is used to construct a generalized form of the Banach limit, an ergodic measure on compact Hausdorff spaces, a stationary extension of a relatively stationary stochastic process xt (0 ≦ t ≦ α) with values in an arbitrary space, and a generalization to arbitrary linear spaces of Krein’s extension theorem for positive-definite complex-valued functions. © 1969 by Pacific Journal of Mathematics.