Egoroff's Theorem Research Articles

Some notes on monotone set-valued measures and Egoroff's theorem

In this note, we point out that Theorem 3 (a version of Egoroff's theorem for monotone set-valued measures) shown in the paper “Lusin's theorem for monotone set-valued measures on topological spaces” (Fuzzy Sets and Systems 364 (2019) 111-123) is not valid, and we present two revised versions. We also point out that the proofs of Lemma 2 and Theorem 1 are defective, thus the truth of their conclusions cannot be affirmed. Thereby, an open problem for the characteristics of monotone measures is raised.

Egoroff's theorems for random sets on non-additive measure spaces

<abstract> We present four versions of Egoroff's theorems for measurable closed-valued multifunctions on non-additive measure spaces. The conditions provided for each of these four versions are not only sufficient, but also necessary. In our discussions the continuity of non-additive measures is not required. The previous related results are improved and generalized. </abstract>

Generalized convergence theorems for monotone measures

In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

Average Cost Optimality Inequality for Markov Decision Processes with Borel Spaces and Universally Measurable Policies

We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model with a Lyapunov-type stability character, we introduce a set of new conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, our result does not require any compactness and continuity conditions on the MDPs. Instead, the main idea is to use the almost-uniform-convergence property of a pointwise convergent sequence of measurable functions as asserted in Egoroff's theorem. Our conditions are formulated in order to exploit this property. Among others, we require that for each state, on selected subsets of actions at that state, the state transition stochastic kernel is majorized by finite measures. We combine this majorization property of the transition kernel with Egoroff's theorem to prove the ACOI.

Lusin's Theorem for monotone set-valued measures on topological spaces

On a topological space, the inner regularity, outer regularity and regularity of a monotone set-valued measure are considered. Their properties are discussed. Based on these, the versions of Egoroff's Theorem and Lusin's Theorem for monotone set-valued measures on a topological space are shown, respectively.

Egoroff Theorem for Operator-Valued Measures in Locally Convex Cones

Egoroff Theorem for Operator-Valued Measures in Locally Convex Cones

Asymptotically Optimal Discrete-Time Nonlinear Filters From Stochastically Convergent State Process Approximations

We consider the problem of approximating optimal in the Minimum Mean Squared Error (MMSE) sense nonlinear filters in a discrete time setting, exploiting properties of stochastically convergent state process approximations. More specifically, we consider a class of nonlinear, partially observable stochastic systems, comprised by a (possibly nonstationary) hidden stochastic process (the state), observed through another conditionally Gaussian stochastic process (the observations). Under general assumptions, we show that, given an approximating process which, for each time step, is stochastically convergent to the state process, an approximate filtering operator can be defined, which converges to the true optimal nonlinear filter of the state in a strong and well defined sense. In particular, the convergence is compact in time and uniform in a completely characterized measurable set of probability measure almost unity, also providing a purely quantitative justification of Egoroff's Theorem for the problem at hand. The results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating optimal nonlinear filters, such as approximate grid based techniques, known to perform well in a variety of applications.

A new necessary and sufficient condition for the Egoroff theorem in non-additive measure theory

This paper shows that a newly defined condition, called condition (M), is a necessary and sufficient condition for the Egoroff theorem in non-additive measure theory. The existing necessary and sufficient conditions for the Egoroff theorem are described by a doubly-indexed sequence of measurable sets, while condition (M) is described by a singly-indexed sequence of measurable sets.

On Lusin's theorem for non-additive measures that take values in an ordered topological vector space

Lusin's theorem was established for real-valued monotone measures under an equivalent condition to Egoroff's theorem recently. In this paper, we show that the same result remains valid for non-additive measures that take values in an ordered topological vector space. We apply our result to several ordered topological vector spaces.

Note on the Regularity of Nonadditive Measures

We consider the regularity for nonadditive measures. We prove that the non-additive measures which satisfy Egoroff's theorem and have pseudometric generating property possess Radon property (strong regularity) on a complete or a locally compact, separable metric space.

Pseudo-Convergences of Sequences of Measurable Functions on Monotone Multimeasure Spaces

In this paper, we study different types of pseudo-convergences of se- quences of measurable functions with respect to set-valued non-additive monotonic set functions and we establish some pseudo-versions of Egoroff theorem in the set-valued case. We also characterize important structural properties of monotone multimeasures. Mathematics Subject Classication

Lusin's theorem on monotone measure spaces

In this paper, a kind of regularity of monotone measures is shown by using the pseudometric generating property of set function (for short (p.g.p.)) and an equivalence condition for Egoroff's theorem. Lusin's theorem, which is well-known in classical measure theory, is generalized to monotone measure spaces. The continuity from above and below of set functions is not required in the discussions, therefore the previous results obtained by Jiang et al. are generalized. An approximation of measurable functions by continuous functions on monotone measure spaces is presented.

A set-valued Egoroff type theorem

In this paper, an Egoroff type theorem for a fuzzy multimeasure defined on a σ -algebra of subsets of an abstract space and taking values in the family of all nonvoid, closed subsets of a real normed space is established. Some necessary and/or sufficient conditions under which Egoroff's theorem is valid are also presented.

On sufficient conditions for the Egoroff theorem of an ordered topological vector space-valued non-additive measure

In this paper, we consider an ordered vector space endowed with a locally full topology, which is called an ordered topological vector space. We show that the Egoroff theorem remains valid for the ordered topological vector space-valued non-additive measure in the following four cases. The first case is that the measure is strongly order totally continuous; the second case is that the measure is strongly order continuous and possesses an additional continuity property suggested by Sun in 1994 when the ordered topological vector space has a certain property; the third case is that the measure is continuous from above and below when the topology is locally convex; the fourth case is that the measure is uniformly autocontinuous from above, continuous from below and strongly order continuous when the topology is locally convex. Our results are applicable to several ordered topological vector spaces.

On sufficient condition for the Egoroff theorem of an ordered vector space-valued non-additive measure

In this paper, we show that the Egoroff theorem remains valid for any ordered vector space-valued non-additive measure if the measure is multiple continuous from above and continuous from below.

An approximation for a subclass of the Riemann-Hilbert problems

Consider the problem of solving a Riemann-Hilbert problem with 'zero index'. Abraham (2000, IMA J. Appl. Math., 65, 257-281) suggested to replace a possibly complicated kernel of a homogeneous Riemann-Hilbert problem with a Pade approximant that uniformly approximates the original kernel. Abraham's procedure fails whenever the kernel cannot be approximated uniformly by a Pade approximant (see Example 1). This article (i) provides an approximation technique to approximate solutions of a non-homogeneous Riemann-Hilbert problem with zero index in L p (R) (1 < p < ∞) sense, which improves the result by Abraham in two directions (weaker conditions on approximating functions and solutions for a non-homogeneous Riemann-Hilbert problem with zero index). Also, we discussed an interesting case p = oo (uniformly approximation). (ii) Using the Egoroff's theorem provides a pointwise approximate solutions for a class of non-homogeneous Riemann-Hilbert problem with zero index. (iii) Using the Shannon sampling theorem provides explicit solutions for certain non-homogeneous Riemann-Hilbert problems with zero index. Some approximations which exploiting this fact will be discussed. (iv) Applications to integral equations are given.

Egoroff's Theorem

In this paper n, k are natural numbers, X is a non empty set, and S is a σ-field of subsets of X. Next we state several propositions: (1) Let M be a σ-measure on S, F be a function from N into S, and given n. Then {x ∈ X: ∧ k (n ≤ k ⇒ x ∈ F (k))} is an element of S. (2) Let F be a sequence of subsets of X and n be an element of N. Then (the superior set sequence of F )(n) = ⋃ rng(F ↑ n) and (the inferior set sequence of F )(n) = ⋂ rng(F ↑ n). (3) Let M be a σ-measure on S and F be a sequence of subsets of S. Then there exists a function G from N into S such that G = the inferior set sequence of F and M(lim inf F ) = sup rng(M ·G).

On the Independence of a Generalized Statement of Egoroff's Theorem from ZFC after T. Weiss

We consider a generalized version (GES) of the wellknown Severini-Egoroff theorem in real analysis, first shown to be undecidable in ZFC by Tomasz Weiss. This independence is easily derived from suitable hypotheses on some cardinal characteristics of the continuum like b and o, the latter being the least cardinality of a subset of [0,1] having full outer measure.

The Egoroff property and the Egoroff theorem in Riesz space-valued non-additive measure theory

The Egoroff theorem remains valid for any Riesz space-valued non-additive measure which is strong order continuous and possesses a form of continuity called “property (S)” in the literature, whenever the Riesz space has the Egoroff property. This version of the Egoroff theorem is also valid for any non-additive measure with the property of uniform autocontinuity, strong order continuity and continuity from below by assuming only the weak σ -distributivity which is weaker than the Egoroff property.

The Egoroff theorem for non-additive measures in Riesz spaces

The Egoroff theorem remains valid for any Riesz space-valued non-additive measure which is continuous from above and below by assuming that the Riesz space has the asymptotic Egoroff property. This property is satisfied for many concrete Riesz spaces, such as R S of all real functions on a non-empty set S, L 0 [ 0 , 1 ] of all Lebesgue measurable functions, and their ideals.