Calculus of Real and Complex Variables

5.5 Measures And Outer Measures

There is also something called an outer measure which is defined on the set of all subsets.

Definition 5.5.1 Let Ω be a nonempty set and let λ : P

→ [0,∞) satisfy the following:

  1. λ
    = 0
  2. If A ⊆ B, then λ
    ≤ λ
  3. λ
    ≤∑ i=1∞λ

Then λ is called an outer measure.

Every measure determines an outer measure. For example, suppose that μ is a measure on ℱ a σ algebra of subsets of Ω. Then define

This is easily seen to be an outer measure. Also, we have the following Proposition.

Proposition 5.5.2 Let μ be a measure as just described. Then

as defined above, is an outer measure and also, if E ∈ℱ, then
= μ
.

Proof: The first two properties of an outer measure are obvious. What of the third? If any

= ∞, then there is nothing to show so suppose each of these is finite. Let Fi ⊇ Ei such that Fi ∈ℱ and
+
> μ
. Then

Since ε is arbitrary, this establishes the third condition. Finally, if E ∈ℱ,  then by definition,
≤ μ
because E ⊇ E. Also, μ
≤ μ
for all F ∈ℱ such that F ⊇ E. If follows that μ
is a lower bound of all such μ
and so
≥ μ
.■

Part August 25, 2018 I  Preliminary Topics

Chapter 1 Basic Notions

1.1 Sets And Set Notation
1.2 The Schroder Bernstein Theorem
1.3 Equivalence Relations
1.4 Well Ordering And Induction
1.5 The Complex Numbers And Fields
1.6 Polar Form Of Complex Numbers
1.7 Roots Of Complex Numbers
1.8 The Quadratic Formula
1.9 The Complex Exponential
1.10 The Cauchy Schwarz Inequality
1.11 Polynomials
1.12 The Fundamental Theorem Of Algebra
1.13 Some Topics From Analysis
1.14 lim sup and lim inf
1.15 Exercises

Chapter 2 Basic Topological and Algebraic Considerations

2.1 Closed and Open Sets
2.2 Compactness
2.2.1 Continuous Functions
2.2.2 Convergent Sequences
2.2.3 Continuity and the Limit of a Sequence
2.2.4 The Extreme Value Theorem and Uniform Continuity
2.2.5 Convergence of Functions
2.2.6 Multiplication of Series
2.2.7 The Distance to a Set
2.3 Tietze Extension Theorem
2.4 Root Test
2.5 Equivalence Of Norms
2.6 Norms On Linear Maps
2.7 Connected Sets
2.8 Stone Weierstrass Approximation Theorem
2.8.1 The Bernstein Polynomials
2.8.2 The Case Of Compact Sets
2.8.3 The Case Of A Closed Set in ℝn
2.8.4 The Case Of Complex Valued Functions
2.9 The Jordan Curve Theorem∗
2.10 Brouwer Fixed Point Theorem∗
2.10.1 Simplices And Triangulations
2.10.2 Labeling Vertices
2.10.3 The Brouwer Fixed Point Theorem
2.11 Exercises

Part II  Real Analysis

Chapter 3 The Derivative, A Linear Transformation

3.1 Basic Definitions
3.2 The Chain Rule
3.3 The Matrix Of The Derivative
3.4 Existence Of The Derivative, C1 Functions
3.5 Mixed Partial Derivatives
3.6 Implicit Function Theorem
3.7 Invariance Of Domain∗
3.8 Exercises

Chapter 4 Line Integrals

4.1 Existence and Definition
4.1.1 Change of Parameter
4.1.2 Existence
4.1.3 The Riemann Integral
4.2 Estimates and Approximations
4.2.1 Finding the Length of a C1 Curve
4.2.2 Curves Defined in Pieces
4.2.3 A Physical Application, Work
4.3 Conservative Vector Fields
4.4 Orientation
4.5 Exercises

Chapter 5 Measures And Measurable Functions

5.1 Measurable Functions
5.2 Measures And Their Properties
5.3 Dynkin’s Lemma
5.4 Measures And Regularity
5.5 Measures And Outer Measures
5.6 Exercises
5.7 An Outer Measure On P (ℝ)
5.8 Measures From Outer Measures
5.9 When Is A Measure A Borel Measure?
5.10 One Dimensional Lebesgue Measure
5.11 Exercises

Chapter 6 The Abstract Lebesgue Integral

6.1 Definition For Nonnegative Measurable Functions
6.1.1 Riemann Integrals For Decreasing Functions
6.1.2 The Lebesgue Integral For Nonnegative Functions
6.2 The Lebesgue Integral For Nonnegative Simple Functions
6.3 The Monotone Convergence Theorem
6.4 Other Definitions
6.5 Fatou’s Lemma
6.6 The Integral’s Righteous Algebraic Desires
6.7 The Lebesgue Integral, L1
6.8 The Dominated Convergence Theorem
6.9 Product Measures
6.10 Some Important Convergence Theorems
6.10.1 Eggoroff’s Theorem
6.10.2 The Vitali Convergence Theorem
6.11 Exercises

Chapter 7 Basic Function Spaces

7.1 Bounded Continuous Functions
7.2 Compactness in C (K, ℝn)
7.3 The Lp Spaces
7.4 Approximation Theorems
7.5 A Useful Inequality
7.6 Uniform Convexity Of Lp
7.7 Exercises

Chapter 8 The Lebesgue Integral In ℝp

8.1 p Dimensional Lebesgue Measure And Integrals
8.2 Lebesgue Measure And Linear Transformations
8.3 Change of Variables for Nonlinear Maps
8.4 Mappings Which Are Not One To One
8.5 Spherical Coordinates In p Dimensions
8.6 Approximation with Smooth Functions
8.7 Continuity Of Translation
8.8 Separability
8.9 Exercises

Chapter 9 Differentiation

9.1 The Besicovitch Covering Theorem
9.2 Fundamental Theorem Of Calculus For Radon Measures
9.3 Vitali Coverings
9.4 Differentiation Of Radon Measures
9.5 The Radon Nikodym Theorem For Radon Measures
9.6 Change of Variables Again
9.7 Exercises

Chapter 10 Some Fundamental Functions and Transforms

10.1 Gamma Function
10.2 Laplace Transform
10.3 Fourier Transform
10.4 Fourier Transforms in ℝn
10.5 Fourier Transforms Of Just About Anything
10.5.1 Fourier Transforms in G∗
10.5.2 Fourier Transforms of Functions In L1 (ℝn)
10.5.3 Fourier Transforms of Functions In L2 (ℝn)
10.5.4 The Schwartz Class
10.5.5 Convolution
10.6 Exercises

Chapter 11 Degree Theory, An Introduction

11.1 Sard’s Lemma and Approximation
11.2 An Identity and Surprising Implications
11.3 Definitions And Elementary Properties
11.4 Borsuk’s Theorem
11.5 Applications
11.6 The Product Formula and Jordan Separation Theorem
11.7 Jordan Curve Theorem
11.8 Exercises

Chapter 12 Green’s Theorem

12.1 An Elementary Form of Green’s Theorem
12.2 Stoke’s Theorem
12.3 A General Green’s Theorem
12.4 Green’s Theorem for a Rectifiable Jordan Curve∗
12.5 Orientation of a Jordan Curve

Part III  Abstract Analysis

Chapter 13 Banach Spaces

13.1 Theorems Based On Baire Category
13.1.1 Baire Category Theorem
13.1.2 Uniform Boundedness Theorem
13.1.3 Open Mapping Theorem
13.1.4 Closed Graph Theorem
13.2 Basic Theory of Hilbert Spaces
13.3 Hahn Banach Theorem
13.3.1 Partially Ordered Sets
13.3.2 Gauge Functions And Hahn Banach Theorem
13.3.3 The Complex Version Of The Hahn Banach Theorem
13.3.4 The Dual Space And Adjoint Operators
13.4 Exercises

Chapter 14 Representation Theorems

14.1 The Lebesgue Decomposition
14.2 Vector Measures
14.3 The Dual Space of Lp (Ω)
14.4 The Dual Space Of L∞(Ω)
14.5 Representations for Positive Linear Functionals on Cc (ℝp)
14.6 Representing a Radon Measure by Slicing
14.7 The Dual Space Of C0 (ℝp)
14.8 Exercises

Part IV  Complex Analysis

Chapter 15 Fundamentals

15.1 Banach Spaces
15.2 The Cauchy Riemann Equations
15.3 Contour Integrals
15.4 Primitives and Cauchy Goursat Theorem
15.5 Functions Differentiable on a Disk, Zeros
15.6 The General Cauchy Integral Formula
15.7 Riemann sphere
15.8 Exercises

Chapter 16 Isolated Singularities and Analytic Functions

16.1 Open Mapping Theorem for Complex Valued Functions
16.2 Functions Analytic on an Annulus
16.3 The Complex Exponential and Winding Number
16.4 Cauchy Integral Formula for a Cycle
16.5 An Example of a Cycle
16.6 Isolated Singularities
16.7 The Residue Theorem
16.8 Evaluation of Improper Integrals
16.9 The Inversion of Laplace Transforms
16.10 Exercises

Chapter 17 Mapping Theorems

17.1 Meromorphic Functions
17.2 Meromorphic on Extended Complex Plane
17.3 Rouche’s Theorem
17.4 Fractional Linear Transformations
17.5 Some Examples
17.6 Riemann Mapping Theorem
17.6.1 Montel’s Theorem
17.6.2 Regions with Square Root Property
17.7 Exercises

Chapter 18 Spectral Theory of Linear Maps

18.1 The Resolvent and Spectral Radius
18.2 Functions of Linear Transformations
18.3 Invariant Subspaces
A.1 Systems of Equations
A.2 Matrices
A.3 Subspaces and Spans
A.4 Application to Matrices
A.5 Mathematical Theory of Determinants
A.5.1 The Function sgn
A.5.2 Determinants
A.5.3 Definition of Determinants
A.5.4 Permuting Rows or Columns
A.5.5 A Symmetric Definition
A.5.6 Alternating Property of the Determinant
A.5.7 Linear Combinations and Determinants
A.5.8 Determinant of a Product
A.5.9 Cofactor Expansions
A.5.10 Formula for the Inverse
A.5.11 Cramer’s Rule
A.5.12 Upper Triangular Matrices
A.6 Cayley-Hamilton Theorem
A.7 Eigenvalues and Eigenvectors of a Matrix
A.7.1 Definition of Eigenvectors and Eigenvalues
A.7.2 Triangular Matrices
A.7.3 Defective and Nondefective Matrices
A.7.4 Diagonalization
A.8 Schur’s Theorem
A.9 Hermitian Matrices
A.10 Right Polar Factorization
A.11 Direct Sums
A.12 Block Diagonal Matrices
B.1 The Hamel Basis
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