Σ-additive Measure Research Articles

Strong 0-1 laws in finite model theory

AbstractWe introduce a new framework for asymptotic probabilities of sentences, in which we have a σ-additive measure on the sample space of all sequencesA= {} of finite models, where the universe ofis {1,2, …, n}. and use this framework to strengthen 0-1 laws for logics.

Measures on diffeomorphism groups for non-archimedean manifolds: Group representations and their applications

Nondegenerate σ-additive measures with ranges in ℝ and ℚq (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚq, where ℚq is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory.

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions.

Choquet integrals and natural extensions of lower probabilities

Both the Choquet integral with respect to monotone set functions and the natural extensions of lower probabilities are generalizations of the Lebesgue integral with respect to σ-additive measures. The relation between these generalizations is investigated. We show that the Choquet integral with respect to a belief measure is always greater than or equal to the corresponding natural extension. Also, we compare some concepts introduced in the theory of imprecise probabilities with concepts established in fuzzy measure theory, capacity theory, and Dempster-Shafer theory.

Doob-meyer decomposition for set-indexed submartingales

Set-indexed martingales and submartingales are defined and studied. The admissible function of a submartingale is defined and some class (D) conditions are given which allow the extension of the function to a σ-additive measure on the predictable σ-algebra. Then, we prove a Doob-Meyer decomposition: A set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. A hypothesis of predictability ensures the uniqueness of this decomposition. An explicit construction of the increasing process associated with a submartingale is given. Finally, some remarks, about quasimartingales are discussed.

Separability of the L1-space of a vector measure

Let Σ be a σ-algebra of subsets of some set Ω and let μ:Σ→[0,∞] be a σ-additive measure. If Σ(μ) denotes the set of all elements of Σ with finite μ-measure (where sets equal μ-a.e. are identified in the usual way), then a metric d can be defined in Σ(μ) by the formulahere E ΔF = (E\F) ∪ (F\E) denotes the symmetric difference of E and F. The measure μ is called separable whenever the metric space (Σ(μ), d) is separable. It is a classical result that μ is separable if and only if the Banach space L1(μ), is separable [8, p.137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If Σ happens to be the σ-algebra of μ-measurable sets constructed (via outer-measure μ*) by extending μ defined originally on merely a semi-ring of sets Γ ⊆ Σ, then it is also classical that the countability of Γ guarantees the separability of μ and hence, also of L1(μ), [8, p. 69].

On the extension of bimeasures

We prove a necessary and sufficient condition for the existence of an extension of a scalar bimeasure on abstract sets to a Σ-additive measure on the generated Σ-algebra. We also prove some extension theorems for vector bimeasures.

Measures on infinite-dimensional orthomodular spaces

We classify the measures on the lattice ℒ of all closed subspaces of infinite-dimensional orthomodular spaces (E, Ψ) over fields of generalized power series with coefficients in ℝ. We prove that every σ-additive measure on ℒ can be obtained by lifting measures from the residual spaces of (E, Ψ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on ℒ is not separating.

On the “zero-two” law for positive contractions

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

Nuclear subspace of L0 and the Kernel of a linear measure

Let E be a locally convex space. Then E is nuclear metrizable if and only if there exists a σ-additive measure μ on E′ such that L: E → L 0( E′, μ), L( x) = 〈 x, ·〉, is an isomorphism. Let E be quasi-complete or barrelled. Suppose that there exists a σ-additive measure ν on E satisfying ( E′, τ ν )′ ⊃ E. Then E′ b is an isomorphic subspace of L 0( E, ν) and nuclear, where b is the strong dual topology and τ ν is the L 0( E, ν) topology. In the case where E is an LF space, for a random linear functional L: E → L 0(Ω, U , P), the next conditions are equivalent: (a) The cylinder set measure μ on E′ determined by L is σ-additive and (b) x n → 0 in E implies that L( x n ) → 0, P-a.s.

Topological Boolean rings. Decomposition of finitely additive set functions

As a basis for the whole paper we establish an isomorphism between the lattice Ίfls(R) of all abounded monotone ring topologies on a Boolean ring R and a suitable uniform completion of R; it follows that Wls(R) itself is a complete Boolean algebra. Using these facts we study 5-bounded monotone ring topologies and topological Boolean rings (conditions for completeness and metriziability, decompositions). In the second part of this paper we give a simple proof of a Lebesgue-type decomposition for finitely additive (e.g. semigroup-valued) set functions on a ring, which was first proved by Traynor (in the group-valued case) answering a question of Drewnowski. Using the Lebesgue-decomposition various other decompositions are obtained. 0. Introduction. The first part of this paper (Chapter 2) deals with an examination of the lattice Ψts(R) of all ^-bounded monotone ring topologies (= FN-topologies) on a Boolean ring R. In (2.2) an isomorphism is established between Wls(R) and the completion of R for the finest ^-bounded monotone ring topology Us on R. From this result we get some consequences (criteria for completeness and metrizability, decomposition theorems for monotone ring topologies) which are also interesting for measure theory and which — as far as they are known in special cases — were before in each case proved with quite different methods. In the second part of this paper (Chapter 3) a decomposition μ — Σa(ΞA μa into an infinite sum is given for an ^-bounded content μ: R -> G defined on a Boolean ring with values in e.g. a complete Hausdorff topological group (content = finitely additive set function); this decomposition includes the usual decomposition theorems as special cases. For an illustration of the method Chapter 3 first deals with the case \A\— 2, i.e. with the Lebesgue decomposition; this again includes as special cases decompositions of a Hewitt-Yosida type, decompositions into an atomless and an atomic content, into a regular and an antiregular content and others. We explain the arising problems with the Lebesgue decomposition. The classic Lebesgue decomposition {μ — λ + v, λ J_w, v < u) of a nonnegative (σ-additive) measure on a σ-algebra rests on a decomposition of the basic set into two disjoint sets of the σ-algebra. The same is still true

On Weak Convergence of Hypermeasures

Measures on the hyperspace of the closed sets with the FLACHSMEYER-FELL topology are completely defined by their capacities. A necessary and sufficient condition is given for the weak convergence of a sequence of positive bounded σ-additive measures on the hyperspace in terms of their capacities.

On states on the product of logics

We take up the question of when a state (= σ-additive measure) on the product of logics (=σ-orthomodular posets) depends on at most countably many coordinates. We show that it is always so provided there are no real-measurable cardinals. The manner of dependence is a kind of convex combination. We derive some consequences of the latter statement.

Characterization of the closed convex hull of the range of a vector-valued measure

For a set K in a locally convex topological vector space X there exists a set T, a σ-algebra S of subsets of T and a σ-additive measure m: S → X such that K is the closed convex hull of the range { m( E): E ∈ S } of the measure m if and only if there exists a conical measure u on X so that K K u , K u , the set of resultants of all conical measures v on X such that v < u.

Konstruktion Subjektiver Wahrscheinlichkeiten1

This paper contains recent results on the construction of subjective probability measures which are almost or strictly agreeing with the qualitative (subjective) probability strueture. The aproaches discussed are exclusively based on Boolean structures which endowed with a binary order relation (‘qualitative probability’) generate a qualitative probability structure. the ropresentation problem can then be formulated as follows: under which conditions (necessary and sufficient) do there exist finitely additive, σ-additive and conditional probability measures compatible with an appropriate qualitative probability structure and in addition satisfying well-known axioms of probability. It is shown that for the solution of the representation problems we can effectively use methods of extensive measurement permitting a behavioristie interpretation. Finally, some remarks concern the role of subjective probability within Bayesian statisties, in partieular regaeding its applications to econometries.