Product Σ-algebra Research Articles

The one-way Fubini property and conditional independence: An equivalence result

A process defined by a continuum of random variables with non-degenerate idiosyncratic risk is not jointly measurable with respect to the usual product σ-algebra. We show that the process is jointly measurable in a one-way Fubini extension of the product space if and only if there is a countably generated σ-algebra given which the random variables are essentially pairwise conditionally independent, while their conditional distributions also satisfy a suitable joint measurability condition. Applications include: (i) new characterizations of essential pairwise independence and essential pairwise exchangeability; (ii) when a one-way Fubini extension exists, the need for the sample space to be saturated if there is an essentially random regular conditional distribution with respect to the usual product σ-algebra.

Marginals and the product strong lifting problem

We study the permanence of strong liftings for products of topological probability spaces. We get results to the positive for the 'usual product' or for the product σ-algebra obtained from it by adjoining the two-sided nil-null sets under assumptions implying that the product topology is contained in this product σ-algebras. For the construction of strong liftings in the product the application of marginals is basic as well as that of the fundamental properties of lifting topologies.

Fubini theorem for non additive measures in the framework of g-expectation

ABSTRACTSince the seminal paper of Ghirardato (1997), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997), Chateauneuf and Lefort (2008) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.

Probability Measures on Product Spaces with Uniform Metrics

For a countable product of complete separable metric spaces, with a topology induced by a uniform metric, the σ-algebra generated by the open balls, which was introduced by Dudley (1966), coincides with the product σ-algebra. Any probability measure on the product space with this σ-algebra is quasi-separable in the sense that, for any union of open balls that has full measure, there is a countable sub-union that also has full measure. With suitably adapted definitions, the topology of weak convergence on the space of such measures is equivalent to the topology induced by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first l factors, l = 1; 2; ...; is a homeomorphism if the range of this mapping is also given a uniform metric. These findings are relevant for the theory of games of incomplete information, where a topology on the space of belief hierarchies that is based on a uniform metric has been proposed as being more appropriate for capturing the continuity properties of strategic behaviour.

Integral Operators on the Product of C(K) Spaces

We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the representing polymeasure of the operator. Some applications are given. In particular, we characterize the Borel polymeasures that can be extended to a measure in the product σ-algebra, generalizing previous results for bimeasures. We also give necessary conditions for the weak compactness of the extension of an integral multilinear operator on a product of C(K) spaces.

Minimal sufficiency of order statistics in convex models

Let P be a convex and dominated statistical model on the measurable space (X ,A), with A minimal sufficient, and let n ∈ N. Then A⊗n sym, the σ-algebra of all permutation invariant sets belonging to the n-fold product σ-algebra A⊗n, is shown to be minimal sufficient for the corresponding model for n independent observations, Pn = {P⊗n : P ∈ P}. The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of A⊗n sym.

On a measure-theoretic problem of Arveson

A probability measure ν \nu on a product space X × Y X\times Y is said to be bistochastic with respect to measures λ \lambda on X X and μ \mu on Y Y if the marginals π 1 ( ν ) \pi _1(\nu ) and π 2 ( μ ) \pi _2(\mu ) are exactly λ \lambda and μ \mu . A solution is presented to a problem of Arveson about sets which are of measure zero for all such ν \nu .

Trace measures of a positive definite bimeasure

If a positive definite bimeasure, M: A × A → C, on any measurable space (Ω, A ) is extendable to a complex measure M′, on the product σ-algebra A ⊗ A , one can define its trace M′ Δ on the diagonal Δ of Ω × Ω. If M is not extendable, the definition of M′ Δ fails, but the need for defining some measures on Δ which would act as M′ Δ remains, especially in the spectral analysis and the analysis of the stationarity of weakly harmonizable processes. In this paper, we fulfill this need with two types of measures concentrated on Δ or, more generally, on simple curves of Ω × Ω. The first ones, which have been called quasi-trace measures, are constructed under the assumption that the given positive definite bimeasure M is σ-finite. They have the expected properties. The second ones, which are called pseudo-trace measures, are constructed, whatever the positive definite bimeasure M be, but they very often lead to new problems.

Equivalent Gaussian measure whose R-N derivative is the exponential of a diagonal form

A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demonstrate the existence of a measurable (relative to the completed product σ-algebra) Gaussian process with covariance K. Using this result, sufficient conditions are given on the means and the covariances (relative to two equivalent (~) Gaussian measures P and P λ ) of a process X so that the Radon-Nikodým (R-N) derivative dp λ dP is the exponential of the diagonal form in X. Analogues of the last two results in the setup of Hilbert space are also proved.

On semi-Markov processes on arbitrary spaces

Let E be an arbitrary set, a σ-algebra of subsets of the Borel sets of F. We write A × B for the product of the sets A and B, andfor the product σ-algebra of and (i.e. the σ-algebra generated by the rectangles A × B with A ∈ and B ∈ ).