Fatou's Lemma Research Articles

Convergence propositions for a generalized daniell-integral

In this paper we shall prove that the extension I of a generalized (or full) Daniell-integral I over a non distributive lattice E with values in a weakly б-distributive 1-group G has all the fundamental limiting properties of the Lebesgue integral, i.e. the properties stated i n the monotone convergence theorem, Fatou's-lemma and Lebesgue's dominated convergence theorem.

A unified approach to several results involving integrals of multifunctions

A well-known equivalence of randomization result of Wald and Wolfowitz states that any Young measure can be regarded as a probability measure on the set of all measurable functions. Here we give a sufficient condition for the Young measure to be equivalent to a probability measure on the set of all integrable selectors of a given multifunction. In this way, Aumann's identity for integrals of multifunctions can be interpreted in a novel fashion. By additionally applying a fundamental result from Young measure theory to uniformlyL1-bounded sequences of functions, Fatou's lemma in several dimensions, which is formulated in terms of the integral of a Kuratowski limes superior multifunction, can be proven in a new fashion. Also, a natural extension of these arguments leads to a generalization of a recent result by Artstein and Rzezuchowski [3].

Méthode de compacité et de décomposition applications: Minimisation, convergence des martingales, lemma de Fatou multivoque

New results of decomposition for bounded sequences in LE1 and in the space of integrably bounded multifunctions with non empty convex weakly compact values in a Banach space E and its applications to problems of Minimization, convergence of martingales, Multivalued Fatou lemma are presented.

On set-valued fuzzy integrals

In the paper, we extend fuzzy integrals of point-valued functions which were defined by Ralescu and Adams to set-valued functions. Some properties similar to Auman integrals are shown, and these include convexity, monotonicity, Fatou's lemma, Lebesgue convergence theorem, etc.

Fatou's Lemmas for the set of integrable selections

In this paper, we prove Fatou-type results for the set of Bochner integrable selections from a set-valued function. Also we obtain some new dominated convergence theorems for the Aumann integral. Our results have applications in Economic Theory.

What do discounted optima converge to?: A theory of discount rate asymptotics in economic models

This paper considers a sequence of stochastic dynamic problems in which the discount rate goes to zero. Sufficient conditions are given under which the limit of discounted optimal policies (and normalized values) are a (i) long-run average optimal and (ii) catching-up optimal policy (and value). These conditions are shown to be satisfied in many economic models. A new decision criterion for the undiscounted problem, the strong long-run average, is introduced. This criterion refines the long-run average, is often equivalent to the catching-up, and is easier to use than the latter since it has a recursive representation. A generalization of Fatou's lemma, which has wider applicability, is also proved.

Convergence of Conditional Expectations for Unbounded Random Sets, Integrands, and Integral Functionals

Given a sequence of unbounded convex random sets, we study under which conditions Fatou's lemma for the weak upper limit of their conditional expectations holds. We also give multivalued versions of dominated and monotone convergence theorems, and we discuss the special case of the integral. Finally, applications to epigraphic convergence of integrands and to Mosco convergence of certain integral functionals are provided.

A counterexample and an exact version of Fatou's lemma in infinite dimensional spaces

A counterexample and an exact version of Fatou's lemma in infinite dimensional spaces

Fatou's lemma in infinite dimensions

By a general method, based on weak convergence of transition probabilities, new infinite-dimensional Fatou lemmas are derived.

More on Fatou's Lemma in Several Dimensions

AbstractRecently, Balder proved a version of FatoiTs lemma in several dimensions which, inter alia, generalizes a version of this lemma due to Artstein. Here we show how the latter result can be used to derive the former, by using Chacon's biting lemma.

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane ( SIAM J. Control 5 (1967) , 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt ( J. Math. Anal. Appl. 7 (1963) , 110–117), C. Olech ( J. Differential Equations 2 (1966) , 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari ( SIAM J. Control 12 (1974) , 319–331) and T. S. Angell ( J. Optim. Theory Appl. 19 (1976) , 63–79).

A General Approach to Lower Semicontinuity and Lower Closure in Optimal Control Theory

A self-contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory, which is based on a central relative weak compactness criterion (called tightness) and relaxation in all but one variable. Two lower closure results for outer integral functionals with variable abstract time domain are developed. The first of these has a convexity condition for the integrand and generalizes all similar results in the literature. The second lower closure result is of a new kind; among other things, it implies a quite general version of Fatou's lemma in several dimensions.

Some remarks on existence of optimal controls for Mayer problems with singular components

This paper proves an existence theorem for optimal controls for systems governed by ordinary differential equations and a large class of functional differential equations of neutral type. Extensions beyond earlier work are made as a result of employing a new closure theorem originally used by Cesari and Suryanarayana in their study of Pareto optima and which is, in turn, based on the Fatou lemma for vector-valued functions as proved by Schmeidler. The use of these techniques simplifies the standard arguments for existence in the presence of singular components and allows the use of very weak semi-normality conditions. It also permits the consideration of a significantly larger class of hereditary systems than has been treated in the existing literature.

The fuzzy integral

In this paper we define the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure. We show that the monotone convergence theorem and Fatou's lemma are still true in this new setting. We study some of the properties of this integral, and show that it coincides with another fuzzy integral defined in the literature. Our main result is a convergence theorem, that is in a way stronger than the Lebesgue-dominated convergence theorem. This holds when the fuzzy measure is also assumed to be subadditive.

A note on fatou's lemma in several dimensions

The theorem answers, in the affirmative, the problem raised by Hildenbrand and Mertens (1971). A partial answer was given in Hildenbrand (1974, lemma 3, p. 69) where it is assumed that {fn(o)},, = r, 2, ,__ is bounded almost everywhere. A need for the full answer arose in the author’s recent work on continuous-time optimal allocation plans (paper forthcoming). We shall employ the theory of integration of correspondences (set-valued functions). Our notations and terminology follow Hildenbrand (1974, ch. I) where all the necessary material can be found; occasionally we shall give a precise reference.

Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, D the real Schwartz space, and D′ its strong dual. D and D′ are partially ordered by C and C′ respectively, where C is the positive cone of nonnegative functions in D and C′ its dual in D′ . C is a strict B -cone and C′ is normal, where B is the family of all bounded subsets of D . If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y( ω) − X( ω) ∈ D′ for almost all ω ∈ Ω(P). Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.

Measure theoretic convergences of observables and operators

Definitions of different types of measure theoretic convergence for observables and operators are given. In particular we define convergence in measure, almost everywhere, everywhere, almost uniformly, and uniformly. These types of convergence are compared and characterized. Furthermore, our theory is compared to that of Segal-Stinespring. Convergence theorems such as a bounded convergence theorem, Fatou's lemma, and a special case of Egoroff's theorem are proved. We show that the general Egoroff's theorem does not hold.

Fatou's Lemma in Several Dimensions

In this note the following generalization of Fatou's lemma is proved: Lemma. Let {fn)n_l be a sequence of integrable functions on a mea- sure space S with values in R+, the nonnegative orthant of a d-dimen- sional Euclidean space, for which ffn—*aGiR+. Then there exists an in- tegrable function f, from S to R+, such that a.e. f(s) is a limit point of VnisV^and ff^a. 1. Introduction. When d = l, the result is a form of Fatou's lemma. The assertion above is applied in mathematical economics (4).2 It is also strongly connected with the theory of set valued functions (2) or correspondences (3). The nontrivial part of the arguments is lim- ited to the case where 5 is an atomless measure space. In the purely atomic case the lemma is reduced to a simple exercise in series. In any case, the lemma cannot be proved by a successive application of Fatou's lemma d times. A few corollaries of the lemma are proved in §3. 2. Preliminary results and the proof of the lemma. Let (A)°=1 be a sequence of (nonempty) subsets of Rd. We denote by Lim Sup^4 the set of all the limit points of the sequences (a),T-i with anEAn, » = 1, 2, • • ■ . Denote by x-y the inner product, Yt=i xlyl, in Rd.

Integration theorems for gages and duality for unimodular groups

affiliated with a. with respect to a on (t. In the present work, we investigate analogues of certain portions of ordinary integration theory which Segal did not have occasion to develop in [10], and then we apply the theory of gage spaces to harmonic analysis and duality of unimodular groups. One of the most important ways of constructing an integral in ordinary measure theory is by starting from a positive linear functional on some algebra of bounded functions. In ?1, the corresponding construction of gages is considered. Theorem 1 gives conditions under which a positive central linear functional on a self-adjoint algebra of bounded operators yields a gage on the W*-algebra it generates. This theorem is applied in ?7 to the construction of product gages and in ?9 to the construction of the dual gage for a unimodular group. The question of what can be said about the extension of noncentral positive linear functionals naturally arises. A special case is dealt with in ?2. In [10], convergence nearly everywhere (n.e.) is defined for gage spaces. In ?3, a notion of convergence in measure is introduced. It is found to have the usual relations to convergence n.e. and mean convergence. In ?4, several dominated convergence theorems are proved together with a version of Fatou's lemma. Convergence in measure is most often used in probability spaces. In later work than [10], Segal has used a notion of convergence in probability for self-adjoint operators on a probability gage space, which is defined differently from our convergence in measure. However, the two are