The Theory of Measures and Integration

[554/47%]

E. M. Vestrup

Chapter 1Set Systems[81/60%]

\pi

\lambda

-Systems, And Semirings[21/100%]

Section 1.2Fields[15/87%]

\sigma

-Fields[21/52%]

Section 1.4The Borel

\sigma

-Field[6/67%]

Section 1.5The k-Dimensional Borel

\sigma

-Field[8/0%]

\sigma

-Fields: Construction and Cardinality

Section 1.7A Class of Ethereal Borel Sets[10/0%]

Chapter 2Measures[39/82%]

Section 2.1Measures[12/92%]

Section 2.2Continuity of Measures[19/89%]

Section 2.3A Class of Measures[5/80%]

Section 2.4Appendix: Proof of the Stieltjes Theorem[3/0%]

Chapter 3Extensions of Measures[51/78%]

Section 3.1Extensions and Restrictions[2/100%]

Section 3.2Outer Measures[5/100%]

Section 3.3Carathéodory's Criterion[7/100%]

Section 3.4Existence of Extensions[7/100%]

Section 3.5Uniqueness of Measures and Extensions[7/71%]

Section 3.6The Completion Theorem[3/100%]

Section 3.7The Relationship Between

\sigma(\mathcal{A})

\mathcal{M}\left(\mu^{*}\right)

[1/100%]

Section 3.8Approximations[5/40%]

Section 3.9A Further Description of

\mathcal{M}\left(\mu^{*}\right)

[10/80%]

Section 3.10A Correspondence Theorem[4/0%]

Chapter 4Lebesgue Measure[58/28%]

Section 4.1Lebesgue Measure: Existence and Uniqueness[9/100%]

Section 4.2Lebesgue Sets

Section 4.3Translation Invariance of Lebesgue Measure[11/64%]

Section 4.4Linear Transformations[10/0%]

Section 4.5The Existence of non-Lebesgue Sets[7/0%]

Section 4.6The Cantor Set and the Lebesgue Function[12/0%]

Section 4.7A Non-Borel Lebesgue Set[3/0%]

Section 4.8The Impossibility Theorem[1/0%]

Section 4.9Excursus: “Extremely Nonmeasurable Sets”[5/0%]

Chapter 5Measurable Functions[49/61%]

Section 5.1Measurability[14/71%]

Section 5.2Combining Measurable Functions[5/100%]

Section 5.3Sequences of Measurable Functions[2/100%]

Section 5.4Almost Everywhere[3/67%]

Section 5.5Simple Functions[4/75%]

Section 5.6Some Convergence Concepts[12/50%]

Section 5.7Continuity and Measurability[7/0%]

Section 5.8A Generalized Definition of Measurability[2/100%]

Chapter 6The Lebesgue Integral[49/65%]

Section 6.1Stage One: Simple Functions[9/100%]

Section 6.2Stage Two: Nonnegative Functions[16/69%]

Section 6.3Stage Three: General Measurable Functions[14/71%]

Section 6.4Stage Four: Almost Everywhere Defined Functions[10/20%]

Chapter 7Integrals Relative to Lebesgue Measure[25/4%]

Section 7.1Semicontinuity[6/17%]

Section 7.2Step Functions in Euclidean Space[2/0%]

Section 7.3The Riemann Integral, Part One[1/0%]

Section 7.4The Riemann Integral, Part Two[11/0%]

Section 7.5Change of Variables in the Linear Case[5/0%]

L^{p}

Spaces[70/44%]

\boldsymbol{L}^{\boldsymbol{p}}

Space: The Case

\mathbf{1} \leqslant \boldsymbol{p}<+\infty

[10/90%]

Section 8.2The Riesz-Fischer Theorem[5/100%]

L^{p}

Space: The Case

0<p<1

[8/50%]

L^{p}

Space: The Case

p=+\infty

[7/43%]

Section 8.5Containment Relations for

\boldsymbol{L}^{\boldsymbol{p}}

Spaces[8/63%]

Section 8.6Approximation[10/0%]

Section 8.7More Convergence Concepts[16/31%]

Section 8.8Prelude to the Riesz Representation Theorem[4/0%]

Section 8.9The Riesz Representation Theorem[2/0%]

Chapter 9The Radon-Nikodym Theorem[43/30%]

Section 9.1The Radon-Nikodym Theorem, Part I[21/62%]

Section 9.2The Radon-Nikodym Theorem, Part II[3/0%]

Section 9.3From Radon-Nikodym to Riesz Representation[16/0%]

Section 9.4Martingale Theorems[3/0%]

Chapter 10Products of Two Measure Spaces[64/16%]

Section 10.1Product Measures[12/67%]

Section 10.2The Fubini Theorems[15/0%]

Section 10.3The Fubini Theorems in Euclidean Space[11/0%]

Section 10.4The Generalized Minkowski Inequality[5/40%]

Section 10.5Convolutions[9/0%]

Section 10.6The Hardy-Littlewood Theorems[12/0%]

Chapter 11Arbitrary Products of Measure Spaces[25/28%]

Section 11.1Notation and Conventions[5/100%]

Section 11.2Construction of the Product Measure[6/33%]

Section 11.3Convergence Theorems in Product Space[9/0%]

Section 11.4The

L^{2}

Strong Law[5/0%]

Section 11.5Prelude to the

L^{1}

Section 11.6The

L^{1}