Sequences Of Measurable Functions Research Articles

Spaces of measurable functions on the Levi-Civita field

We introduce the ℒp spaces of measurable functions whose pth power is summable with respect to the uniform measure over the Levi-Civita field. These spaces are the counterparts of the real Lp spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the Lp spaces: for instance, the ℒp spaces are not sequentially complete with respect to the p-norm. This motivates the study of the completions of the ℒp spaces with respect to strong convergence, denoted by ℒsp. It turns out that the ℒsp spaces are Banach spaces and that it is possible to define an inner product over ℒs2, thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in ℛ that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over ℛ is consistent with the result obtained with other nonlinear generalized functions.

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

In this note, the convergence of the sum of two convergent sequences of measurable functions is studied by means of two types of absolute continuity of fuzzy measures, i.e., strong absolute continuity of Type I, and Type VI. The discussions of convergence a.e. and convergence in measure are done in the general framework relating to a pair of monotone measures, and general results are shown. The previous related results are generalized.

The pseudo-convergence of measurable functions on set-valued fuzzy measure space

For sequences of measurable functions on a set-valued fuzzy measure space, the concepts of pseudo almost everywhere convergence, pseudo almost uniformly convergence, and pseudo-convergence in measure are introduced. Then, Egoroff’s theorem, Lebesgue’s theorem, and Riesz’s theorem are generalized from real-valued fuzzy measure spaces onto set-valued fuzzy measure spaces.

New results on integration on the Levi-Civita field

New results for integration of functions on the Levi-Civita field R are presented in this paper which is a continuation of the work done in Shamseddine and Berz (2003) [13] and complements it. For example, we show that if f and g are bounded on a measurable set A and f=g almost everywhere on A then f is measurable on A if and only if g is measurable on A in which case the integrals of f and g over A are equal. We also show that if A⊂R is measurable and if (fn) is a sequence of measurable functions that converge uniformly on A to f, then f itself is measurable on A and its integral over A is given by ∫Af=limn→∞∫Afn.

CONVERGENCE OF GENERALIZED SUGENO INTEGRALS ON QUASI-ADDITIVE MEASURE SPACE

On quasi-additive measure space,the generalized Sugeno integral is defined again by introducing the induced operator and the quasi-multiplication operator.For the sequence of measurable functions which converges in quasi-additive measure,the convergence of generalized Sugeno integrals is obtained by means of the operation of the quasi-additive and quasi-multiplication operator.Finally,the monotone increasing convergence theorem of this kind of integrals is given.

Onp-Convergence in Measure of a Sequence of Measurable Functions

In the study by Papanastassiou and Papachristodoulos, 2009 the notion ofp-convergence in measure was introduced. In a natural wayp-convergence in measure induces an equivalence relation on the spaceMof all sequences of measurable functions converging in measure to zero. We show that the quotient spaceℳis a complete but not compact metric space.

Selection principle for pointwise bounded sequences of functions

For a number ɛ > 0 and a real function f on an interval [a, b], denote by N(ɛ, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [ai, bi], i = 1, …, n, on [a, b] such that |f(ai) − f(bi)| > ɛ for any i = 1, …, n (sup O = 0). The following theorem is proved: if {fj} is a pointwise bounded sequence of real functions on the interval [a, b] such that n(ɛ) ≡ lim supj→∞N(ɛ, fj, [a, b]) 0, then the sequence {fj} contains a subsequence which converges, everywhere on [a, b], to some function f such that N(ɛ, f, [a, b]) ≤ n(ɛ) for any ɛ > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {fj} and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.

Improvement of an Estimate of the Global Existence Theorem for Solutions of the Bogoliubov Equations

For a one-dimensional system of particles interacting like elastic balls, we improve an estimate of the existence theorem for the global solution of the Cauchy problem for the Bogoliubov equations with initial data from the space of sequences of measurable functions.

Lp,q-bounded sequences in function spaces

A sequence of measurable functions {fn} is called lp,q-bounded sequence, 0<p<∞, 0<q≤∞, if for any sequence of real numbers a={αn}∈lp,q we have sup⁡n⁡|αnFn(ω)|<∞ ω-a.e., for any sequence {Fn} such that for every n functions fn anf Fn are equimeasurable. The main result gives necessary and sufficient conditions for the sequence being lp,q-bounded.

Convergence for Sequences of Functions and an Egorov Type Theorem

For every ordinal ξ<ω1 we define a new type of convergence for sequences of functions (ξ-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this type of convergence we obtain an Egorov type theorem for sequences of measurable functions.

Purely Game-Theoretic Random Sequences: II. Limiting Empirical Distributions and Strong Central Limit Theorem

In Part I of this paper [M. Minozzo, Theory Probab. Appl., 44 (1999), pp. 511--522] a definition of typical sequences was given, without using any Kolmogorovian probability distribution P, by applying the principle of the excluded gambling strategy directly to a sequence of measurable functions. In this paper we forward this theory by deriving for these typical sequences some elementary limiting empirical distribution functions and some strong central limit theorem type results for the coin tossing process.

A common unique fixed point theorem for two random operators in Hilbert spaces

We construct a sequence of measurable functions and consider its convergence to the unique common random fixed point of two random operators defined on a nonempty closed subset of a separable Hilbert space. The corresponding result in the nonrandom case is also obtained.

CONVERGENCE THEOREMS FOR SEQUENCES OF CHOQUET INTEGRALS

The Choquet integral with respect to nonadditive monotone set functions, including imprecise probabilities and fuzzy measures, is a generalization of the classical Lebesgue integral. It is one kind of nonlinear functionals defined on a subspace of all real valued measurable functions. In this paper, several different types of convergence, including the mean convergence that is based on the Choquet integral, for sequences of measurable functions are considered, and the corresponding convergence theorems for sequence of Choquel integrals are demonstrated. Particularly, the theorem of convergence in measure is presented in a form of “necessary and sufficient condition” by using the structural characteristics of nonnegative monotone set functions. As an application of convergence theorems, the stability of a class of nonlinear integral systems is discussed.

Sequences of measurable functions

Sequences of measurable functions

Convergence almost everywhere of sequences of measurable functions

Convergence almost everywhere of sequences of measurable functions

Some Pointwise Convergence Results in Lp(μ), 1 &lt; p &lt; ∞

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

Pointwise in Terms of Weak Convergence

Let $(\Omega ,\mathfrak {F},\mu )$ be a measure space, $\mu (\Omega ) < \infty$. Let ${X_n}$ be a sequence of measurable functions on $\Omega$ taking values in a compact metric space $M$. The set of bounded stopping times $\tau$ for the ${X_n}$ is a directed set under the obvious ordering. The following theorem is proved: ${X_n}$ converges pointwise almost everywhere if and only if the generalized sequence $\int {\phi ({X_\tau })d\mu }$ converges for every continuous function $\phi$ on $M$. The martingale theorem is proved as an application.

Functions defined by sequences of integrals and the inversion of approximate derived numbers

have been determined.t It is easy to construct sequences for which the limit on the left side of (1) exists and is different from the right side. In the present paper necessary and sufficient conditions are obtained for the existence of this limit in terms of the sequence s,(x). It turns out that the limit function F(x) is independent of f(x). In fact if F(x) is an arbitrary continuous function, there exists a sequence of summable functions sn(x) tending to zero everywhere, for which <sn(x)dx tends to F(x) everywhere. If Fl(x), F2(x), is a sequence of measurable functions, there is a sequence sn(x) tending to zero everywhere for which almost everywhere the set of limits of f sn(x)dx is the sequence Fl(x), F2(x), I f . . fIf esn(x)dx is bounded in n and e, e any measurable subset of (a, b), then F(x), when it exists, is of bounded variation on (a, b). Conversely, if F(x) is of bounded variation, there exists a function f(x) and a sequence of summable functions s (x) tending everywhere to f(x) for which f:Sn(x)dx tends to F(x), and for which fesndx is bounded in n and e. This is of some interest for the reason that it provides a characterization of functions of bounded variation which can be extended to functions of any number of variables. It is possible for the limit of the left side of (1) to exist when the function f(x) is not summable. As an aid in the study of this situation we introduce the following conventions:

Some Uniqueness Theorems for Functions of Class L p

Let {Φ n (x)}, n=1, 2, . . . , be a sequence of measurable functions of a real variable x defined almost everywhere in an interval a≤x≤b. Let f(x) be a function belonging to the Lebesgue class L p (p>0) and let {Φ n } be such that all the integrals c n (f)=∫f(x)Φ n (x)dx (1) exist as L-integrals.