A representation theorem for measures on infinite dimensional spaces

Franz Harpain and Maurice Sion

Vol. 30 (1969), No. 1, 47–58

DOI: 10.2140/pjm.1969.30.47

Abstract

If X is a locally compact, regular topological space, then the well known Riesz representation theorem sets up an isomorphism between the family of all bounded Radón outer measures on X and the set of continuous positive linear functionals on the family of continuous functions with compact support in X. In this isomorphism corresponding elements, l a linear functional and μ a measure, satisfy the relationship l(f) = ∫ fdμ for all continuous functions f with compact support in X.

Since an infinite product of locally compact, regular spaces is in general no longer locally compact with respect to the product topology, the Riesz representation theorem fails to hold for such spaces. In this paper, an analogue of the Riesz representation theorem is obtained for this case.

Mathematical Subject Classification

Primary: 28.40

Milestones

Received: 22 October 1968

Published: 1 July 1969