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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]