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Let G be a family of subsets of an n-element set. The family G is called non-trivial 3-wise intersecting if the intersection of any three subsets in G is non-empty, but the intersection of all subsets is empty. For a real number p ∈ (0 , 1) we define the measure of the family by the sum of p | G | (1 - p) n - | G | over all G ∈ G . We determine the maximum measure of non-trivial 3-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems. [ABSTRACT FROM AUTHOR]

It is proved that the family of measurable real functions on the product measure space satisfying (or not) the conclusion of Fubini's Theorem (along with a number of related families) is algebraically large, in the sense that it contains large convex cones or even large vector subspaces (except for zero) under rather general assumptions. Several earlier related results are improved, mainly regarding the cardinality of the generator set of the corresponding cones or subspaces. [ABSTRACT FROM AUTHOR]

A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions. The study of schemes using arbitrary probability spaces and unbounded number of participants allows us to investigate their abstract properties, to connect the topic to other branches of mathematics, and to discover new design paradigms. A scheme is perfect if unqualified subsets have no information on the secret, that is, their total share is independent of the secret. By relaxing this security requirement, three other scheme types are defined. Our first result is that every (infinite) access structure can be realized by a perfect scheme where the recovery functions are non-measurable. The construction is based on a paradoxical pair of independent random variables which determine each other. Restricting the recovery functions to be measurable ones, we give a complete characterization of access structures realizable by each type of the schemes. In addition, either a vector-space or a Hilbert-space based scheme is constructed realizing the access structure. While the former one uses the traditional uniform distributions, the latter one uses Gaussian distributions, leading to a new design paradigm. [ABSTRACT FROM AUTHOR]

We consider the family 𝒞 ⁢ 𝒮 {\mathcal{CS}} of symmetric Cantor subsets of [ 0 , 1 ] {[0,1]}. Each set in 𝒞 ⁢ 𝒮 {\mathcal{CS}} is uniquely determined by a sequence a = (a n) {a=(a_{n})} belonging to the Polish space X : = (0 , 1) ℕ {X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in 𝒞 ⁢ 𝒮 {\mathcal{CS}} and sequences in X. If 𝒜 ⊂ 𝒞 ⁢ 𝒮 {\mathcal{A}\subset\mathcal{CS}} , the corresponding subset of X is denoted by 𝒜 ∗ {\mathcal{A}^{\ast}}. We study the subfamilies ℋ 0 {\mathcal{H}_{0}} , 𝒮 ⁢ 𝒫 {\mathcal{SP}} and ℳ {\mathcal{M}} of 𝒞 ⁢ 𝒮 {\mathcal{CS}} , consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have ℳ ⊂ ℋ 0 ⊂ 𝒮 ⁢ 𝒫 {\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}} , and these inclusions are proper. We prove that the sets ℳ ∗ {\mathcal{M}^{\ast}} , ℋ 0 ∗ {\mathcal{H}_{0}^{\ast}} , 𝒮 ⁢ 𝒫 ∗ {\mathcal{SP}^{\ast}} are residual in X, and μ ⁢ (ℋ 0 ∗) = 0 {\mu(\mathcal{H}_{0}^{\ast})=0} , μ ⁢ (𝒮 ⁢ 𝒫 ∗) = 1 {\mu(\mathcal{SP}^{\ast})=1}. [ABSTRACT FROM AUTHOR]

Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini's theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini's theorem (30) was proved in complete form by this article. [ABSTRACT FROM AUTHOR]

de Finetti's Dutch book theorem explains why probability has to be additive on disjunctions of incompatible yes-no (Boolean) events. The theorem holds verbatim also for continuous events, or random variables, as formalized in Łukasiewicz logic. Independence, a subtler notion than incompatibility, has a probability-free definition: two events are logically independent if there is no logical relation between them. Our main result in this paper is that product is the only coherence preserving operation on books on logically independent sets of continuous (as well as yes-no) events. Thus, also the product law for stochastically independent sets of events is implicit in de Finetti's deeper notion of a coherent betting system. [ABSTRACT FROM AUTHOR]

In this paper, we consider sufficient and necessary condition for the inclusion between Lorentz spaces with mixed norms for two different product measures on (X × Y;A × B): Later, we discuss the Wiener-Ditkin sets of Lorentz spaces with mixed norms and Lorentz spaces. [ABSTRACT FROM AUTHOR]

Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p ≥ − 1 , we prove that the product X × Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1 / ( 1 + p − 1 ) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X × Y when Y satisfies a Prékopa–Leindler inequality. In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in R n (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality. Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results. [ABSTRACT FROM AUTHOR]

We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti's notion of coherence. [ABSTRACT FROM AUTHOR]

The motivation for extending secret sharing schemes to cases when either the set of players is infinite or the domain from which the secret and/or the shares are drawn is infinite or both, is similar to the case when switching to abstract probability spaces from classical combinatorial probability. It might shed new light on old problems, could connect seemingly unrelated problems, and unify diverse phenomena. Definitions equivalent in the finitary case could be very much different when switching to infinity, signifying their difference. The standard requirement that qualified subsets should be able to determine the secret has different interpretations in spite of the fact that, by assumption, all participants have infinite computing power. The requirement that unqualified subsets should have no or limited information on the secret suggests that we also need some probability distribution. In the infinite case events with zero probability are not necessarily impossible, and we should decide whether bad events with zero probability are allowed or not. In this paper, rather than giving precise definitions, we enlist an abundance of hopefully interesting infinite secret sharing schemes. These schemes touch quite diverse areas of mathematics such as projective geometry, stochastic processes and Hilbert spaces. Nevertheless our main tools are from probability theory. The examples discussed here serve as foundation and illustration to the more theory oriented companion paper [arXiv:1310.7423].

The purpose of this article is to show Fubini's theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini's theorem, and prove the relevant proposition. Finally we show Fubini's theorem on measure. [ABSTRACT FROM AUTHOR]

In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the multifractal structure of the thick points for the product measure of two independent stable subordinators' occupation measures. Furthermore, we get the Hausdorff dimension of the thick points. [ABSTRACT FROM AUTHOR]

Let G=( V, E) be a locally finite graph. Let $\vec{p}\in[0,1]^{V}$. We show that Shearer's measure, introduced in the context of the Lovász Local Lemma, with marginal distribution determined by $\vec{p}$, exists on G if and only if every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionally, we derive a non-trivial uniform lower bound for the parameter vector of the dominated Bernoulli product field. This generalises previous results by Liggett, Schonmann, and Stacey in the homogeneous case, in particular on the k-fuzz of ℤ. Using the connection between Shearer's measure and a hardcore lattice gas established by Scott and Sokal, we transfer bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem. [ABSTRACT FROM AUTHOR]

In the paper we present two results for measures on projections in a W-algebra of type I. First, it is shown that, for any such measure m, there exists a Hilbert-valued orthogonal vector measure µ such that |µ( p)| = m( p) for every projection p. In view of J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803-806), this means that the above assertion is valid for an arbitrary W-algebra. Secondly, a construction of a product measure on projections in a W-algebra of type I (an analogue of the product measure in classical Lebesgue theory) is proposed. [ABSTRACT FROM AUTHOR]

It is shown that well-orderings are either negligible or nonmeasurable with respect to the completions of σ-finite product measures. Several consequences of this fact are discussed in the light of some classical problems formulated by Hilbert, Lebesgue and Luzin. [ABSTRACT FROM AUTHOR]

For non-additive set functions, the independent product, in general, is not unique and the Fubini theorem is restricted to slice-comonotonic functions. In this paper, we use the representation theorem of Gilboa and Schmeidler (Math Oper Res 20:197-212, ) to extend the Möbius product for non-additive set functions to non-finite spaces. We extend the uniqueness result of Ghirardato (J Econ Theory 73:261-291, ) for products of two belief functions and weaken the requirements on the marginals necessary to obtain the Fubini property in the product. More importantly, we show that for the Möbius product one side of the Fubini theorem holds for all integrable functions if one of the marginals either is a probability or a convex combination of a chain of unanimity games, i.e., we relax the requirement of slice-comonotonicity and enrich the set of possible applications. [ABSTRACT FROM AUTHOR]

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km ink for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non- homogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument, illustrated in a simulation appendix and extended to (non-homogeneous) products of log-concave measures. [ABSTRACT FROM AUTHOR]

Here we give a positive answer to the so-called Product Measure Problem under the relatively simple hypothesis that the measure in one of the factors is inner regular and its support with the induced Hausdorff topology is locally metrizable. No special hypothesis on the other topological measure space is required. The proof is inspired by the rather imprecise conjecture that the open sets of a topological space must satisfy some restrictions in order to support any strictly positive σ-finite measure. [ABSTRACT FROM AUTHOR]

In this article, we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure μ being defined on the set of all configurations which are given on a graph G. We show that if μ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators. [ABSTRACT FROM AUTHOR]