The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform

Michel Théra - One of the best experts on this subject based on the ideXlab platform.

Van Ngai Huynh - One of the best experts on this subject based on the ideXlab platform.

Jun Kawabe - One of the best experts on this subject based on the ideXlab platform.

  • MDAI - Convergence in Measure Theorems of the Choquet Integral Revisited.
    Modeling Decisions for Artificial Intelligence, 2020
    Co-Authors: Jun Kawabe
    Abstract:

    The validity of the monotone Convergence theorem, the Fatou and the reverse Fatou lemmas, and the Dominated Convergence theorem of the Choquet integral of measurable functions converging in measure are fully characterized by the conditional versions of the monotone autocontinuity and the autocontinuity. In those theorems the nonadditive measure may be infinite and the functions may be unbounded. The dual measure forms and the extension to symmetric and asymmetric Choquet integrals are also discussed.

  • The Vitali Convergence in measure theorem of nonlinear integrals
    Fuzzy Sets and Systems, 2020
    Co-Authors: Jun Kawabe
    Abstract:

    Abstract The Vitali Convergence in measure theorem for the abstract Lebesgue integral is fundamental in Lebesgue integration theory and yields the bounded Convergence theorem and the Dominated Convergence theorem as its applications. In this paper, in a unified way using the perturbation method the Vitali Convergence in measure theorem is established for nonlinear integrals such as the Choquet, Sipos, Sugeno, and Shilkret integrals, and their symmetric and asymmetric extensions. It is derived from the Fatou and the reverse Fatou type lemmas for perturbative nonlinear integral functionals.

  • The Choquet integral in Riesz space
    Fuzzy Sets and Systems, 2008
    Co-Authors: Jun Kawabe
    Abstract:

    A comprehensive discussion of the theory of Choquet integration in a Riesz space is given. In particular, it is proved that the monotone Convergence theorem, the Fatou lemma, and the Dominated Convergence theorem are still valid for Riesz space-valued non-additive measures if we assume that the Riesz space has a new property concerning the cardinality of the set of points of discontinuity of a monotone function.

Milan Tvrdý - One of the best experts on this subject based on the ideXlab platform.

  • Bounded Convergence theorem for abstract Kurzweil–Stieltjes integral
    Monatshefte für Mathematik, 2016
    Co-Authors: Giselle Antunes Monteiro, Umi Mahnuna Hanung, Milan Tvrdý
    Abstract:

    In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the $$\sigma $$ σ -Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

  • Bounded Convergence theorem for abstract Kurzweil-Stieltjes integral
    Monatshefte für Mathematik, 2015
    Co-Authors: Giselle Antunes Monteiro, Umi Mahnuna Hanung, Milan Tvrdý
    Abstract:

    In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzela or Arzela–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the $$\sigma $$ -Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzela Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

Zeng-tai Gong - One of the best experts on this subject based on the ideXlab platform.