Robert Gardner's Online Real Analysis 1

Online Real Analysis 1
with online videos Dr. Bob's Online University
(Not an actual university)
Robert 'Dr. Bob' Gardner

This website includes links to class notes and supplements (in PDF) used in the teaching of East Tennessee State University's Real Analysis 1 (MATH 5210). Links are also available for video presentations of the notes. The videos have been used as of the teaching of online Real Analysis 1, starting in fall 2020. The material is largely based on H. L. Royden and P.M. Fitzpatrick Real Analysis, 4th Edition, Pearson (2010):

Royden and Fitzpatrick's Real Analysis book, 4th edition

Robert "Dr. Bob" Gardner can be reached by e-mail at: gardnerr@etsu.edu. His office is in Gilbreath Hall, Room 308F on the ETSU campus.

The following notes and supplements are all in PDF. The videos can be streamed from YouTube and should remain permanently available. (The videos previously on ETSU's Panopto host site have been, or may soon be, deleted.)

Background and Motivation.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
Introduction to Math Philosophy and Meaning	Intro and Philosophy	-	-
Essential Background for Real Analysis I	Essential Background	-	Background Video (1:26:32)
The Riemann-Lebesgue Theorem	The Riemann-Lebesgue Theorem	-	Riemann Lebesgue Video, Part 1 (34:37) Riemann Lebesgue Video, Part 2 (51:52) Riemann Lebesgue Video, Part 3 (56:26)

Chapter 1. The Real Numbers: Sets, Sequences, and Functions.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
1.4. Open Sets, Closed Sets, and Borel Sets or Real Numbers	1.4 Notes	1.4 Supplement 1.4 Supplement Print File	1.4 Video (54:27)

Chapter 2. Lebesgue Measure.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
2.1. Introduction	2.1 Notes	2.1 Supplement 2.1 Supplement Print File	2.1 Video (24:39)
2.2. Lebesgue Outer Measure	2.2 Notes	2.2 Supplement 2.2 Supplement Print File	2.2 Video, Part 1 (33:32) 2.2 Video, Part 2 (29:39)
2.3. The σ-Algebra of Lebesgue Measurable Sets	2.3 Notes	2.3 Supplement 2.3 Supplement Print File	2.3 Video, Part 1 (1:07:44) 2.3 Video, Part 2 (44:42)
2.4. Outer and Inner Approximation of Lebesgue Measurable Sets	2.4 Notes	2.4 Supplement 2.4 Supplement Print File	2.4 Video (49:44)
2.5. Countable Additivity, Continuity, and the Borel-Cantelli Lemma	2.5 Notes	2.5 Supplement 2.5 Supplement Print File	2.5 Video (46:11)
Supplement: Dr. Bob's Axiom of Choice Centennial Lecture	Axiom of Choice	-	Axiom of Choice Video (42:28)
2.6. Nonmeasurable Sets (from Royden's 3rd Edition)	2.6 Notes (3rd ed.)	2.6 Supplement (3rd ed.) 2.6 Supplement Print File (3rd ed.)	2.6 3rd Edition Video (51:31)
2.6. Nonmeasurable Sets (from Royden and Fitzpatrick's 4th Edition)	2.6 Notes (4th ed.)	2.6 Supplement (4th ed.) 2.6 Supplement Print File (4th ed.)	2.6 4th Edition Video (26:12)
Supplement: Nonmeasurable Sets and the Banach-Tarski Paradox	Banach-Tarski Paradox	-	Banach-Tarski Video (46:51)
2.7. The Cantor Set and the Cantor Lebesgue Function	2.7 Notes	2.7 Supplement 2.7 Supplement Print File	2.7 Video (1:07:23)
Supplement: The Cardinalities of the Set of Measurable Sets and the Set of Nonmeasurable Sets	Cardinality of ℳ	-	Cardinality of ℳ Video (26:07)

Chapter 3. Lebesgue Measurable Functions.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
3.1. Sums, Products, and Compositions	3.1 Notes	3.1 Supplement 3.1 Supplement Print File	3.1 Video (58:48)
3.2. Sequential Pointwise Limits and Simple Approximation	3.2 Notes	3.2 Supplement 3.2 Supplement Print File	3.2 Video (55:35)
3.3. Littlewood's Three Principals, Egoroff's Theorem, and Lusin's Theorem	3.3 Notes	3.3 Supplement 3.3 Supplement Print File	3.3 Video (59:55)

Chapter 4. Lebesgue Integration.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
4.1. The Riemann Integral	4.1 Notes	-	4.1 Video (14:22)
4.2. The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure	4.2 Notes	4.2 Supplement 4.2 Supplement Print File	4.2 Video (1:37:08)
4.3. The Lebesgue Integral of a Measurable Nonnegative Function	4.3 Notes	4.3 Supplement 4.3 Supplement Print File	4.3 Video (1:50:11)
Supplement: The Dirac Delta Function - A Cautionary Tale	Dirac Delta	-	Dirac Delta Video (24:08)
4.4. The General Lebesgue Integral	4.4 Notes	4.4 Supplement 4.4 Supplement Print File	4.4 Video (57:40)
4.5. Countable Additivity and Continuity of Integration	4.5 Notes	4.5 Supplement 4.5 Supplement Print File	4.5 Video (10:43)
4.6. Uniform Integrability: The Vitali Convergence Theorem	4.6 Notes	4.6 Supplement 4.6 Supplement Print File	4.6 Video (50:42)

Chapter 5. Lebesgue Integration: Further Topics.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
5.1. Uniform Integrability and Tightness: A General Vitali Convergence Theorem	5.1 Notes	5.1 Supplement 5.1 Supplement Print File	5.1 Video (27:57)
5.2. Convergence in Measure	5.2 Notes	5.2 Supplement 5.2 Supplement Print File	5.2 Video (32:45)
5.3. Characterization of Riemann and Lebesgue Integrability	5.3 Notes	5.3 Supplement 5.3 Supplement Print File	5.3 Video (32:12)

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Last updated: March 15, 2024.