Measurability of Lattice Operations in a Cone

Abstract

Abstract.Let A" be a locally convex Hausdorff topological vector space and C a convex cone generating X such that C is a lattice in its own order. Under suitable conditions (x, y)--sup(x, y) and ¡nf(jr, y) are shown to be measurable mappings.Let A' be a locally convex Hausdorff topological vector space over the real numbers. Let C be a closed proper convex cone with vertex 0 and let C generate X. Further, let C be a lattice in its own order. There are wellknown results asserting the continuity of the mappings ix, y>)->-sup(x, y) and ix, y)->-inf(.x, y) under suitable restrictions on the cone C ( [4, Chapter V], [2, Appendix]). In this note we shall give conditions under which the lattice operations are measurable mappings. This measurability was found to be very useful in our recent work in potential theory [1]. Theorem 1. Let X be a Hausdorff locally convex real topological vector space. Let C be a closed proper convex cone with vertex at the origin, generating X and such that C is a lattice in its own order. Let B be a compact metrizable base for C.Then, the mappings: CxC^-C given by ix, y)->-sup(x, y) and ix, y)->-inf(x, y) are Borel, viz., the inverse image of any Borel set of C under each of these mappings is a Borel set ofiCxC. Proof.Step (1). Let us denote by K the set of all positive continuous linear functionals on X, and Y=K-K the vector space generated by this cone. We note that F separates the points of A' [4, Example 25, p. 71] and hence o-(x, y) on A1 is a Hausdorff topology. Let us denote by a the topology induced on C by aix, y) on X. Let t be the given topology on X. We note that (C, t) is locally compact, metrizable and separable and hence it is a polish space. It follows that the (C, a) Borel sets and the (C, t) Borel sets are identical and the same is the case on the product space CxC with the respective product topologies [5].