Measurability of lattice operations in a cone

Abstract

Let X 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 inf(x, y) are shown to be measurable mappings. Let X 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 (x, y) osup(x, y) and (x, y)-4inf(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 suich that C is a lattice in its own order. Let B be a compact metrizable base for C. Then, the mappings: Cx C-C given by (x,y)-*sup(x,y) and (x,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 of C x C. 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 Y separates the points of X [4, Example 25, p. 71 ] and hence a(x, y) on X is a Hausdorff topology. Let us denote by a the topology induced on C by a(x, y) on X. Let r be the given topology on X. We note that (C, r) is locally compact, metrizable and separable and hence it is a polish space. It follows that the (C, a) Borel sets and the (C, r) Borel sets are identical and the same is the case on the product space C x C with the respective product topologies [5]. Presented to the Society, September 18, 1972; received by the editors September I 1, 1972 and, in revised form, March 12, 1973. AMS (MOS) subject classifications (1970). Primary 46A40; Secondary 28A20, 31C05, 31D05.