Irreducible partitions and the construction of quasi-measures

Abstract

Abstract. A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact Hausdorf space. We investigate the nature of irreducible partitions as defined by Aarnes and use the results to construct quasi-measures when g(X) = 1. The cohomology ring is an important tool for this investigation.

Covering spaces and irreducible partitions

“…The result then follows from Proposition 1. This result should be compared to Corollary 18 and Proposition 20 of [3]. There, similar results are obtained with d(X) in place of n(X) where d(X) is the maximal rank of an isotropic subgroup of the cohomology group H 1 (X).…”

“…In [4], a new construction was discovered for the case where X is the torus. In [3], the current author analyzed the "irreducible partitions" in the construction theorem and related them to aspects of the cohomology ring of the underlying space. It is the goal of this paper to show that the fundamental group of the space can be used instead and to generalize the recent construction given by Knudsen in [4] to this more abstract setting.…”

“…We let g(X) + 1 denote the largest number of sides for a closed solid subset of X. In [3], basic facts about the number g(X) are given when X is compact, connected and locally connected. Definition 2.…”

Quasi-states and symplectic intersections

“…Existence of a symplectic quasi-measure, say τ , in this case follows from a work by Grubb (see Theorem 32 of [24], where the auxiliary quasi-measures used in the definition of τ are taken to be the standard Lebesgue measure). The value of τ on any 2-dimensional smooth connected closed submanifold with boundary W ⊂ T 2 can be calculated as follows (see Theorem 32 of [24]). If W is contractible in T 2 we have τ (W ) = 0.…”

Дефектные Топологические Меры И Порождаемые Ими Функционалы

“…Условиям теоремы 24 удовлетворяют, например, шар или сфера в R n , а также любые гомеоморфные им топологические пространства (см. [8], [16]). Далее мы конструируем пространство, на котором проблема тоже решается, хотя, вообще говоря, g(X) ̸ = 0.…”