Vol. 28, No. 1, 1969

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Invariant extensions of linear functionals, with applications to measures and stochastic processes

E. J. McShane, Robert Breckenridge Warfield, Jr. and V. M. Warfield

Vol. 28 (1969), No. 1, 121–142

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 Gn1 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 Gk1 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 a) with values in an arbitrary space, and a generalization to arbitrary linear spaces of Krein’s extension theorem for positive-definite complex-valued functions.

Mathematical Subject Classification
Primary: 46.06
Secondary: 28.00
Received: 7 November 1967
Published: 1 January 1969
E. J. McShane
Robert Breckenridge Warfield, Jr.
V. M. Warfield