Convergence theorems and the Lebesgue integral

2011 ◽  
pp. 53-145
Keyword(s):  
Lebesgue Integral ◽  
Convergence Theorems

Measure Theory and Lebesgue Integration Theories

2015 ◽  
pp. 29-46
Keyword(s):  
Probability Theory ◽  
Measure Theory ◽  
Higher Order ◽  
Order Logic ◽  
Lebesgue Integral ◽  
Real Numbers ◽  
Convergence Theorems ◽  
Higher Order Logic ◽  
Lebesgue Integration

As discussed in the previous chapter, the fundamental theories of measure and Lebesgue integration are the prerequisites for the formalization of probability theory in higher-order logic. The scope of this chapter is primarily the formalization of these foundations. Both formalizations of measure theory and Lebesgue integral (Mhamdi, Hasan, & Tahar, 2011), presented in this chapter, are based on the extended-real numbers. This allows us to define sigma-finite and even infinite measures and handle extended-real-valued measurable functions. It also allows us to verify the properties of the Lebesgue integral and its convergence theorems for arbitrary functions. Therefore, the chapter begins with a description of the formalization of extended real numbers.

STRONG AND Δ - CONVERGENCE THEOREMS USING MODIFIED PROXIMAL POINT ALGORITHM IN CAT(0) SPACES

2018 ◽  
Vol 7 (2) ◽  
pp. 8
Author(s):  
KUMAR DAS APURVA ◽  
DHAR DIWAN SHAILESH ◽  
DASHPUTRE SAMIR ◽  
◽  
◽  
...  
Keyword(s):  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Convergence Theorems

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

Characterization of decomposition integrals extending Lebesgue integral

2021 ◽  
Author(s):  
Jun Li ◽  
Radko Mesiar ◽  
Yao Ouyang ◽  
Adam Šeliga
Keyword(s):  
Lebesgue Integral

scholarly journals A Class of Integral Operators that Fix Exponential Functions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini ◽  
Gumrah Uysal ◽  
Basar Yilmaz
Keyword(s):  
General Class ◽  
Pointwise Convergence ◽  
Type Theorem ◽  
Integral Operators ◽  
Exponential Functions ◽  
Convergence Theorems ◽  
Korovkin Type Theorem ◽  
Type Formula

AbstractIn this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

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