Construction of non-subadditive measures and discretization of Borel measures

Abstract

Abstract. The main result of the paper provides a method for construction of regular non-subadditive measures in compact Hausdorff spaces. This result is followed by several examples. In the last section it is shown that "discretization" of ordinary measures is possible in the following sense. Given a positive regular Borel measure λ, one may construct a sequence of non-subadditive measures µn, each of which only takes a finite set of values, and such that µn converges to λ in the w * -topology. Introduction.In this paper we continue the study of non-subadditive measures undertaken in [1], [2] and [5], called there "quasi-measures". They are set-functions defined on the open and on the closed subsets of a locally compact Hausdorff space X, and represent a genuine generalization of regular Borel measures in such spaces. This paper is devoted to showing how they arise and may be constructed when X is compact, and to giving some applications.Non-subadditive measures (NSA-measures), as the name indicates, are generally not subadditive. Indeed, if they are, then they turn out to be ordinary regular Borel measures. This lack of subadditivity is what makes NSA-measures different, and in some respects more interesting than ordinary measures. Instead of weighing effects or events on an additive scale, the NSA-measures register a cumulative effect of events. To produce a certain result, several other results must occur simultaneously. This is of course a very superficial description, and only future development and applications can substantiate what we indicate here.Even if NSA-measures by definition are generalizations of ordinary measures, their existence is not an obvious matter, and turns out to be closely linked to properties of the underlying topological space. The existence of NSA-measures was first established in the author's paper [1]. In [2] we gave

Quasi-states and symplectic intersections

“…A set A ⊂ S 2 is called solid if both A and S 2 \ A are connected. According to the results of Aarnes [5] and Aarnes and Rustad [6], the quasi-measure τ is completely defined by the following condition: for a closed solid set A ⊂ S 2 one has τ (A) = 1 if the Lebesgue measure of A is greater or equal to 1/2 and τ (A) = 0 otherwise.…”

Probability and quasi-measures - a new interpretation

“…The last equality follows from Lemma 3.1 and 3.1 in [2], which implies that for each compact C & U P o s there is a solid compact set C H & U with C & C H . We have now verified conditions A and B for a solid set-function.…”

“…Their basic construction has been given in [1], [2] and [3]. The main construction result ( [1], Theorem 5.1) assumes that a set function " initially is given on a fundamental family of sets a s , called the solid sets, and extended to all of a (a set A P a is solid if A and its compliment are both connected).…”

New topological measures on the torus

“…On a space which is simply co-connected [4] or of Aarnes genus zero [6], there is a standard procedure for constructing simple topological measures. These measures are the generalized point-measures.…”