On the existence of product stochastic measures

In this paper, we prove that the existence of product stochastic measures depends on the axiom-system of set theory: If one accepts the axiom of choice, the answer is negative, and we give a counter-example where the product stochastic measure doesn't exist; but in the Solovay model (one kind of set theory which refuses the axiom of choice), the answer is positive, and we give a proof.     

A mathematical proof of S. Shelah’s theorem on the measure problem and related results

Recently, S. Shelah proved that an inaccessible cardinal is necessary to build a model of set theory in which every set of reals is Lebesgue measurable. We give a simpler and metamathematically free proof of Shelah's result. As a corollary, we get an elementary proof of the following result (without choice axiom): assume there exists an uncountable well ordered set of reals, then there exists a non-measurable set of reals. We also get results about Baire property,K _σ-regularity and Ramsey property.     

The rigid relation principle,a new weak choice principle

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well‐orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo‐Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.     

On grand Lebesgue spaces on sets of infinite measure

We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so‐called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces.     

How good is lebesgue measure?

So, what is the answer to the question “How good is Lebesgue measure?” In the class of invariant measures, Lebesgue measure seems to be the best candidate to be a canonical measure. In the class of countably additive not necessarily invariant measures, to find a universal measure we have to use a strong additional set-theoretical assumption and this seems to be too high a price. Thus the best improvement of Lebesgue measure seems to be the Banach construction of a finitely additive isometrically invariant extension of Lebesgue measure on the plane and line. However, such a measure does not exist on Rⁿ for n ≤ 3, and to keep the theory of measures uniform for all dimensions we cannot accept the Banach measure on the plane as the best solution to the measure problem. From this discussion it seems clear that there is no reason to depose Lebesgue measure from the place it has in modern mathematics. Lebesgue measure also has a nice topological property called regularity: for every EL and every ɛ > 0, there exists an open set V⊃E and closed set F ⊂ E such that m(V/F) < ɛ. It is not difficult to prove that Lebesgue measure is the richest countably additive measure having this property (see [Ru], Thm. 2.20, p. 50).     

Lebesgue积分几何意义的推广

证明了非负有界函数的Lebesgue上积分等于函数下方图形的Lebesgue外测度,其Lebesgue下积分等于函数下方图形的Lebesgue内测度,从而将积分的几何意义从可测情形推广到一般情形.     

基于半范数的风险测度

由于一致性风险测度公理中的平移不变性公理存在不合理性,故可将该该公理从一致性风险测度公理中去掉.将半范数的概念加以扩展,增加单调性要求,则去掉平移不变性公理之后一致性风险测度公理与半范数的要求就完全相同,这样风险度量从本质上讲就是定义在某空间上的半范数.本文发现F ishburn的风险测度是满足正齐次性、次可加性、单调性要求的.从这个意义讲,F ishburn的风险测度是一个比较科学的风险度量方法.     

Shadows of the axiom of choice in the universe $$L(\mathbb {R})$$

We show that several theorems about Polish spaces, which depend on the axiom of choice (\(\mathcal {AC}\)), have interesting corollaries that are theorems of the theory \(\mathcal {ZF} + \mathcal {DC}\), where \(\mathcal {DC}\) is the axiom of dependent choices. Surprisingly it is natural to use the full \(\mathcal {AC}\) to prove the existence of these proofs; in fact we do not even know the proofs in \(\mathcal {ZF} + \mathcal {DC}\). Let \(\mathcal {AD}\) denote the axiom of determinacy. We show also, in the theory \(\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})\), a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in \(\mathcal {ZFC}\), but assuming the existence of large enough cardinals it is true in the model \(\langle L(\mathbb {R}),\in \rangle \).     

A general axiomatic system for image resolution quantification

In this paper, we generalize our previously published axiom system for quantification of image resolution and prove that any resolution measure consistent with the new axiom system must be a homogeneous symmetric function of order 1/2 of the eigenvalues of the covariance matrix of the PSF. We demonstrate that the previous axiom system is not consistent with the affine transformation axiom. We propose a weak combination axiom to replace the previous strong combination axiom and use it to solve this conflict. It is remarkable that the original finding in one-dimension by Wang and Li can be easily rediscovered with aid of the weak combination axiom, instead of using the previous strong combination axiom. If the previous axiom system is modified with the weak combination axiom and augmented with the affine transformation axiom, the resolution measure is shown to be proportional to the squared root of the geometric mean of the eigenvalues of the covariance matrix of the PSF. Relevant discussions and possible extensions are also provided.     

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.     

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs–Markov–Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.     

A Reconciliation Among Discrete Compromise Solutions

The application of compromise solutions to discrete multi-objective problems brings about some technical flexibilities, such as the selection of distance function for computing both normalized attribute ratings and distances between two alternatives, and the choice between the ideal and negative-ideal alternatives for implementing the axiom of choice. These flexibilities are undesirable, since the method may yield conflicting preference-alternative rankings, depending on parameter choice. This paper introduces a credibility measurement of distance function and takes a broader concept of the axiom of choice in order to reconcile disagreement among compromise solutions.     

关于自保形测度与Lebesgue测度的等价性

本文我们研究了自保形测度与Lebesgu测度的关系,对Yuvla Peres等的结果进行了推广,证明了自相似测度要么是奇异的,要么关于Lebesgue测度 绝对连续的,并且若将Lebesgue测度限制在自相似测度的紧支撑上,则其关于非奇异的自相似测度是绝对连续的。     

Injective power objects and the axiom of choice

“The axiom of choice states that any set X of non-empty sets has a choice function—i.e. a function satisfying f(x)∈x for all x∈X. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in , the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing “non-empty” by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for , then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zorn’s lemma and the well-order principle. The injective form of Zorn’s lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice.     

Uniformly spread measures and vector fields

We show that two different ideas of uniform spreading of locally finite measures on the d-dimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure. Bibliography: 11 titles.     

σ-kompakte Räume

The theorem, that -compact spaces are Lindelöf, is equivalent to the countable axiom of choice. Variants of this theorem are compared with weak versions of the axiom of choice.     

Some equivalents of AC in algebra II

We show that the axiom of choice AC is equivalent to the statement Any quotient group of any abelian group has a selector. We also show that the multiple choice axiom MC is equivalent to the assertion: Any filter in any Boolean ring has a well ordered filterbase. Received October 3, 1996; accepted in final form May 1, 1998.     

Dedekind-Endlichkeit und Wohlordenbarkeit

Weak forms of the axiom of choice are defined, their set theoretical strength is investigated and as an application topological wellordering theorems are derived.     

直观模糊集的熵、距离测度、相似测度及它们的关系

熵、距离测度和相似测度是模糊集理论中的三个重要概念.首先系统地给出了直观模糊集的熵、距离测度和相似测度的公理化定义,并讨论了它们之间的一些基本关系.然后在距离测度公理化定义的基础上产生了一些新的直观模糊集的熵公式.