General Measure Spaces Research Articles

Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions

Some Hardy-type integral inequalities in general measure spaces, where the corresponding Hardy operator is replaced by a more general Volterra type integral operator with kernel k(x,y), are considered. The equivalence of such inequalities on the cones of non-negative respective non-increasing functions are established and applied.

Bisimulation and cocongruence for probabilistic systems

We introduce a new notion of bisimulation, called event bisimulation on labelled Markov processes (LMPs) and compare it with the, now standard, notion of probabilistic bisimulation, originally due to Larsen and Skou. Event bisimulation uses a sub σ-algebra as the basic carrier of information rather than an equivalence relation. The resulting notion is thus based on measurable subsets rather than on points: hence the name. Event bisimulation applies smoothly for general measure spaces; bisimulation, on the other hand, is known only to work satisfactorily for analytic spaces. We prove the logical characterization theorem for event bisimulation without having to invoke any of the subtle aspects of analytic spaces that feature prominently in the corresponding proof for ordinary bisimulation. These complexities only arise when we show that on analytic spaces the two concepts coincide. We show that the concept of event bisimulation arises naturally from taking the co-congruence point of view for probabilistic systems. We show that the theory can be given a pleasing categorical treatment in line with general coalgebraic principles. As an easy application of these ideas we develop a notion of “almost sure” bisimulation; the theory comes almost “for free” once we modify Giry’s monad appropriately.

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals\(\ {\, \int^{\!\!\!\!\!\!e}_{E}} f(u) M_\alpha(du)\) of a deterministic function \(f(u)\ge 0\) with respect to a random \(\alpha-\)Frechet (\(\alpha>0\)) sup-measure. The measure \(M_\alpha\) is sup-additive rather than additive and is defined over a general measure space \((E,{\mathcal E}, \mu)\), where \(\mu\) is a deterministic control measure. The extremal integral is constructed in a way similar to the usual \(\alpha-\)stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary \(f(u)\ge 0,\ \int_E f(u)^\alpha \mu(du) <\infty\), and the metric \(\rho_\alpha(f,g) := \int_E |f(u)^\alpha - g(u)^\alpha| \mu(du)\) metrizes the convergence in probability of the resulting integrals.

The Riemann Integral Using Ordered Open Coverings

We define the Riemann integral for bounded functions defined on a general topological measure space. When the space is a compact metric space the integral is equivalent to the R-integral defined by Edalat using domain theory.

Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces

We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion processes on a locally compact separable metric measure space $(M,\rho , \mu )$. These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.

A simple proof of the discrete Steffensen's inequality

sum_{i=n-k_2+1}^n x_ilesum_{i=1}^n x_iy_ilesum_{i=1}^{k_1} x_i.

Some Convergence Results for p-Harmonic Functions on Metric Measure Spaces

In this note we consider a stability property of p-harmonic functions in general metric measure spaces. Despite the fact that there may not be a corresponding differential equation, it is shown here that a family of uniformly convergent p-harmonic functions converges to a p-harmonic function; that is, the collection of p-harmonic functions forms a closed subspace in a certain Hölder class. As a consequence, it is shown that a family of p-harmonic functions bounded in a Sobolev-type space (called the Newtonian space) has a sequence that converges locally uniformly in the domain of harmonicity to a p-harmonic function. This result is used to construct p-harmonic functions on unbounded domains. We also use this convergence result to prove a characterization of a parabolicity property of metric measure spaces. This characterization has been given for Riemannian manifolds by Holopainen, and the result here is a generalization of Holopainen's result. 2000 Mathematics Subject Classification 31C05 (primary), 31C45, 30C99 (secondary)

Approximation, complete continuity, and uniform measurability of Uryson operators on general measure spaces

Approximation, complete continuity, and uniform measurability of Uryson operators on general measure spaces

Exponential approximations in completely regular topological spaces and extensions of Sanov’s theorem

This paper is devoted to the well known transformations that preserve a large deviation principle (LDP), namely, the contraction principle with approximately continuous maps and the concepts of exponential equivalence and exponential approximations. We generalize these transformations to completely regular topological state spaces, give some examples and, as an illustration, reprove a generalization of Sanov’s theorem, due to de Acosta (J. Appl. Probab. 31 A (1994) 41–47). Using partition-dependent couplings, we then extend this version of Sanov’s theorem to triangular arrays and prove a full LDP for the empirical measures of exchangeable sequences with a general measurable state space.

Dempster Combination Rule for Signed Belief Functions

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

A new class of unbalanced haar wavelets that form an unconditional basis for Lp on general measure spaces

Given a complete separable σ-finite measure space (X,Σ, μ) and nested partitions of X, we construct unbalanced Haar-like wavelets on X that form an unconditional basis for Lp (X,Σ, μ) where1<p<∞. Our construction and proofs build upon ideas of Burkholder and Mitrea. We show that if(X,Σ, μ) is not purely atomic, then the unconditional basis constant of our basis is (max(p, q) −1). We derive a fast algorithm to compute the coefficients.

On the setwise convergence of sequences of measures

We consider a sequence {μn} of (nonnegative) measures on a general measurable space (X, ℬ). We establish sufficient conditions for setwise convergence and convergence in total variation.

An improved Menshov-Rademacher theorem

We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval $[0,1]$ and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.

Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.

Bernstein operators and finite exchangeability

We present a method for proving finite versions of De Finetti-type theorems for general measurable spaces ( X, A ). Bernstein operators on the one dimensional simplex [0, 1] are generalized to Bernstein operators on limits of simplices, representing distributions on ( X, A ). The positivity of finite moment sequences allows the approximation, via the generalized Bernstein operators, of exchangeable or partially exchangeable distributions by mixtures of product measures or mixtures of certain subsets of product measures. Examples, including sequences of spherically invariant variables and sequences of separately rotatable random matrices, are given.

Characterizations of conditional expectation operators for Lsbp-valued functions on a general measure space

Characterizations of conditional expectation operators for Lsbp-valued functions on a general measure space

On a Geometrical Representation of Probability Distributions and its use in Statistical Inference.

ABSTRACT: This paper has been divided into six sections. The first three sections are confined to the consideration of the geometry of a sa.mple space with finite number of points : the first is concerned with the representation of a single probability distribution by a point in a finite dimensional Euclidean space and the geometrical interpretation of the laws of probability theory; the second deals with the relationship between two probability distributions defined on the same space and tbe third gives some results of differential geometry applied to such a space and geometrical representation of Fisher's information matrix. The fourth section indicates bow the concepts can be extended to general probability measures defined on a general measurable space. The fifth section gives an inadequate account of the nse of the concepts developed earlier to problems of statistical inference. The last section is not directly concerned with distance, but contains a numder of allied concepts.

Generalized holomorphic processes and differentiability

Holomorphic processes are studied when the parameter set is a general finite atomless separable measure space, using the differentiation operator and Malliavin calculus. Generalized martingales are connected with differentiable processes. In the two-parameter case, these methods give a new simpler proof of the characterization of holomorphic processes by power series expansion.

Littlewood-Paley theory on Gaussian spaces

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

On non-singular renewal kernels with an application to a semigroup of transition kernels

We study the non-singularity and limit properties of the renewal kernel R=∑K ∗n associated with a positive convolution kernel K( x,d y×d t) defined on a general measurable space ( E, E ). The principal tool is the use of embedded renewal measures. As an application we consider continuous parameter semigroups ( R t ( x,d y); t⩾0) of transition kernels on ( E, E ).