Lebesgue's Convergence Theorem of Complex-Valued Function

In this article, we formalized Lebesgue’s Convergence theorem of complex-valued function. We proved Lebesgue’s Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue’s Convergence Theorem of complexvalued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.Narita Keiko - Hirosaki-city, Aomori, JapanEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008, doi:10.2478/v10037-008-0023-1.Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.Noboru Endou, Yasunari Shidama, and Keiko Narita. Egoroff's theorem. Formalized Mathematics, 16(1):57-63, 2008, doi:10.2478/v10037-008-0009-z.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.Keiko Narita, Noboru Endou, and Yasunari Shidama. The measurability of complex-valued functional sequences. Formalized Mathematics, 17(2):89-97, 2009, doi: 10.2478/v10037-009-0010-1.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

An Analysis of the Theory of Functions of One Real Variable

Few undergraduates are aware that the Riemann integral taught in introductory calculus courses has only limited application-essentially this integral can be used only to integrate continuous functions over intervals. The necessity to integrate a broader class of functions over a wider range of sets that arises in many applications motivates the theory of abstract integration and functional analysis. The founder of this theory was the French mathematician Henri Lebesgue, who in 1902 defined the Lebesgue measure of subsets of the real line. The purpose of this project is to elucidate the theory of abstract measure spaces and of important spaces of functions (a critical example of which are Banach spaces), and extend the application of this theory. Developing the tools for doing so has been the focus of my advisor Professor Dmitry Khavinson and me over the past three years. The primary goal of the thesis is to make this highly formal and abstract material accessible to an undergraduate having only a year of coursework in advanced calculus. These concepts are typically introduced at the graduate level, but the ideas require only a familiarity with the analytic style of proof learned as an undergraduate. It would be advantageous to expose advanced undergraduates to this material since these ideas form the foundation for how mathematical research is done at the professional level. The addition of interesting and practical examples (which are scarce in the standard graduate texts) will help to make the concepts more familiar and down-to-earth. The motivation for a new theory of integration came from the Riemann integral\u27s apparent inability to operate on functions that fail to be continuous. For example, the Riemann integral of the function that assigns the value 1 to rational numbers and 0 to irrational numbers can be evaluated over the interval [0, 1] with equally valid justification to be 0 or 1. This is because the definition of the Riemann integral depends on partitioning the domain of the function to be integrated, and finding the maximum and minimum values of the function over each partition. The Lebesgue integral, on the other hand, partitions the range of the function to be integrated and then considers the length of the Jason Reed and Dmitry Khavinson pre-image of each partition as well as the maximum and minimum values of the function of the partition. The utility of this change of perspective arises when we refine what is meant by length in the aforementioned pre-image. The Riemann integral requires that the domain consist of intervals of real numbers (where length makes sense), while the Lebesgue integral can be used with a much broader class of sets. Lebesgue modified the notion of length by defining the measure of a set E to be the smallest possible total length of all collections of intervals that cover E. Using this ingenious method, Lebesgue constructed a theory of integration which forms the most useful example of all general integration theories. The theory has important applications in many areas of science and engineering as well as probability and statistics. Our approach to the subject has emphasized theory developed in H.L. Royden\u27s classic text, Real Analysis. My project has included analysis of each concept in the text, and I have developed for each major subject a collection of problems solved and applications of major theorems that were explored. The result has been comprehension of many of the foundational ideas in the field. We have used a number of supplemental texts to gain depth of understanding where Royden\u27s text provides only a survey, such as the Riesz Representation theorem, and to extend important ideas, such as the consideration of complex-valued (in addition to real-valued) measures. The synthesis has been a comprehensive paper which describes the theoretical directions the research has taken, the major results and theorems with proof, and applications and examples which are worked out in detail. The final record of my research will be divided into the following six sections: Lebesgue measure, Lebesgue integral, relationship between differentiation and Lebesgue integration, Banach space theory, abstract measure theory, and general integration theory. The analysis encompasses discussion of the main ideas (what it means for a set function to be a measure, how an integral can be defined in a coherent way with respect to a measure, when the derivative of an integral of a function is the function itself, different ideas about what it means for a sequence of functions to converge to a function, what are the properties of Banach spaces and why they are useful, etc.), as well as important ideas and theorems that interrelate these concepts (i.e., when we can interchange the limit of a sequence of functions and the integral, how we can represent a bounded linear functional, the structure of certain spaces of integrable functions)

Distributions: The evolution of a mathematical theory

AbstractThe theory of distributions, or generalized functions, evolved from various concepts of generalized solutions of partial differential equations and generalized differentiation. Some of the principal steps in this evolution are described in this paper

Almost periodic functions and dini's theorem

No abstract

Harmonic functions

In this thesis, we study the following topics in complex analysis:- (a) Some basic results like Lebesegue’s covering lemma, Maximum modulus theorem and Schwarz lemma. (b)The basic principle of Normal Family; results like Roche’s theorem, Hurwitz theorem and the Montel’s theorem. (c) The basic theory regarding Harmonic functions. Moreover, in this thesis we plan to focous on the advance theory of Harmonic functions. We study Poisson kernel and it’s properties and finally we give the detail prove of the Harnack’s theorem