Inspired by the “two envelopes exchange paradox,” a finitely additive probability measure m on the natural numbers is introduced. The measure is uniform in the sense that m({i})=m({j}) for all i,j ∈N. The measure is shown to be translation invariant and has such desirable properties as m({i∈N | i≡ 0(mod2)})= 1/2. For any r ∈ [0,1], a set A is constructed such that m(A)= r ; however, m is not de...

We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich’s theorem saying that every proper analytic subgroup of the r...

Misclassification and measurement error are very common in studies of effects of exposures and are often ignored. The problem of exposure misclassification for interaction measures in the context of both multiplicative interaction (1–4) and additive interaction (5) has been discussed previously. Other investigators have discussed problems of outcome misclassification for a single exposure (6, 7...

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density which can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs type measure on a growing interval with ...

Fuzzy measures can flexibly describe the relative importance of decision criterion as well as their interactions in multicriteria decision making. Based on the diamond pairwise comparison, a new identification method of 2-order additive fuzzy measure is proposed. The relative weight and the interaction degree can be obtained simultaneously for every pair of criteria in the diamond pairwise comp...

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel c...

We present a combinatorial description of those families 5a of sets, for which there is a finite measure p. such that inf{ß{P) : P 6 a"} > 0 . This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions. The main part of this paper deals with the following problem. Given a ...

We extend in a natural way Szemerédi’s Regularity Lemma to abstract measure spaces. 1 Introduction In this note we extend Szemerédi’s Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to …nd general conditions under which the original proof of Szemerédi still works. Another extension of SRL to probality spaces was proved by Tao [3], but his results do not imply our most general...