All Integrals
Author: Zhiming Ou
Mattermatics Learning Center
Publisher: 3265 Public Way
ISBN: 978-1-0677109-5-8
Summary
Calculus studies functions with “good” properties, such as continuity, differentiability, integrability. The cardinality of continuous functions is equal to the continuum c; but the set of real valued functions f: ℝ → ℝ has a cardinality of 2c, that is, the cardinality of the power set of continuous functions. Can we calculate the mean values for all real-valued functions?
Lebesgue developed a different kind of integral. In Rieman Integrals, one partition the domain [a, b] into subintervals and sum up the function values over the subintervals. Lebesgue partitioned the range of the function, multiplied by the measure of the corresponding independent variables, and then summed up.
This book summarized all known kinds of integrals, such as Rieman-Stiltjes integral, Feynman integral, etc. We still need a united definition for all integrals; and also, the integral should be related to the “derivative” of some function. Also, there is no formula for the reciprocal of a complicated integrand. Is there an integral transform so that its reciprocal can be expressed as an integral? Study needs to continue.
Content
Chapter 1 Set Theory 5
- 1.1 Concepts 6
Operations on Sets
Construction of Sets 8
- 1.2 Structures of Sets 9
Ordered Sets1
Minima/Maxima, Infimum and Supremum
Cardinality
Countability
Relation between Two Sets
- 1.3 Set of Real Numbers 13
Dedekind Cut and Dedekind’s Principle
The Archimedean Principle
Cantor’s Principle
Weierstrass’ Principle
Infimum and Supremum
Cauchy’s Principle
Bolzano-Weierstrass Principle
Heine-Borel Theorem (finite open cover)
Representation of a real number in base p 17
The Continuous Ordinal Number c, Cantor’s Theorem
Chapter 2 Measure Theory 19
- 1 Set of Points in R1
Closed, Open and Compact Subsets
Cantor’s Ternary Set
- 2 Lebesgue Measure in R1 22
Outer Measure, Properties of the Outer Measure 23
Inner Measure 25
Caratheodory Condition 27
Approximating Measurable Sets
Properties of the Lebesgue Measure
Other Measures: Hausdorff measure, Probability measure.
Chapter 3 Sets of Points in a Metric Space 32
- 1 Point Sets in the Complex Plane 32
- 2 Complete Sets 36
Concept of distance, convergence by distance
Fixed-Point Theorem 39
- 3 Compact Sets 40
Sequentially Compact Set 42
Dense Set 42
Compact Set 42
Continuous Mapping 43
- 4 Distances between Subsets 44
Chapter 4 Riemann-Stieltjes Integrals 45
- 1 Functions of Bounded Variation 46
- 2 Riemann–Stieltjes Integral 48
2.1 Definition
2.2 Existence of RS-Integral 49
Darboux Sums
2.3 Relation to the Riemann integral
2.4 Properties
Chapter 5 Measurable Functions 52
- 5.1 Definition
Positive part and negative part of a function
Simple Functions 57
- 5.2 Egorov Theorem: Uniform Convergence 58
Luzin's version 60
Korovkin's version
- 3 Lusin’s Theorem: Continuity of a Measurable Function 62
- 4 Convergence by Measure 63
Riesz Theorem 63
Chapter 6 Lebesgue Integral 64
- 1 Review of Riemann Integral
Cauchy’s Criteria for Riemann Integrability 65
Convergence for Riemann Integrable functions 66
- 2 Lebesgue Integral 67
- 3 Approach by Simple Functions 70
Young’s Approach: Similar to Riemann Integral
Lebesgue Integral for General Measurable Function 74
- 4 Convergence Theorems 77
Chapter 7 Generalized Derivatives 80
- 1 Derivatives of a Function 80
Dini Derivatives 81
Lebesgue’s Differentiation Theorem
Functions of Bounded Variation
Absolutely Continuous Functions
- 2 Fundamental Theorems of Lebesgue Integrals 85
Other Fundamental Theorems 89
Chapter 8 Schrodinger’s Equation and The Feynman Integral 92
- 1 Probability amplitude 92
- 2 Feynman’s Path Integral 95
- 3 Solution from Functional analysis 97
- 4 my solution
Chapter 9 Transforms of Integrals 97
- 1 Laplace Transform 97
- 2 Fourier Transform 101
- 3 Mellin Transform 103
Unsolved problems in the theory for integrals 105