Robert Gardner's Online Real Analysis 1

Online Real Analysis 1
with videos and transcript on the ETSU faculty server Dr. Bob's Online University
(Not an actual university)

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):

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 are mp4 and the transcripts are ASCII files (readable with Microsoft Notepad, for example; these were generated by Zoom and are not precise). For videos which stream from Zoom, see Online Real Analysis 1 with Zoom videos (the Zoom videos are not controlled by ETSU and may be taken down at some point).

Background and Motivation.

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

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

SECTION	NOTES	SUPPLEMENTS	VIDEOS	TRANSCRIPTS
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)	1.4 Transcript

Chapter 2. Lebesgue Measure.

SECTION	NOTES	SUPPLEMENTS	VIDEOS	TRANSCRIPTS
2.1. Introduction	2.1 Notes	2.1 Supplement 2.1 Supplement Print File	2.1 Video (24:39)	2.1 Transcript
2.2. Lebesgue Outer Measure	2.2 Notes	2.2 Supplement 2.2 Supplement Print File	2.2 Video, Part 1 (33:33) 2.2 Video, Part 2 (29:39)	2.2 Transcript, Part 1 2.2 Transcript, Part 2
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)	2.3 Transcript, Part 1 2.3 Transcript, Part 2
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)	2.4 Transcript
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)	2.5 Transcript
Supplement: Dr. Bob's Axiom of Choice Centennial Lecture	Axiom of Choice	-	Axiom of Choice Video (42:28)	Axiom of Choice Transcript
2.6. Nonmeasurable Sets (from Royden and Fitzpatrick's 3rd Edition)	2.6 Notes (3rd ed.)	2.6 Supplement (3rd ed.) 2.6 Supplement Print File (3rd ed.)	2.6 Video, 3rd Edition (52:31)	2.6 3rd Edition Transcript
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 Video, 4th Edition (26:12)	2.6 Video, 4th Edition
Supplement: Nonmeasurable Sets and the Banach-Tarski Paradox	Banach-Tarski Paradox	-	Banach-Tarski Video (46:51)	Banach-Tarski Transcript
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)	2.7 Transcript
Supplement: The Cardinalities of the Set of Measurable Sets and the Set of Nonmeasurable Sets	Cardinality of ℳ	-	Cardinality of ℳ Video (26:07)	Cardinality of ℳ Transcript

Chapter 3. Lebesgue Measurable Functions.

SECTION	NOTES	SUPPLEMENTS	VIDEOS	TRANSCRIPTS
3.1. Sums, Products, and Compositions	3.1 Notes	3.1 Supplement 3.1 Supplement Print File	3.1 Video (58:48)	3.1 Transcript
3.2. Sequential Pointwise Limits and Simple Approximation	3.2 Notes	3.2 Supplement 3.2 Supplement Print File	3.2 Video (55:35)	3.2 Transcript
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)	3.3 Transcript

Chapter 4. Lebesgue Integration.

SECTION	NOTES	SUPPLEMENTS	VIDEOS	TRANSCRIPTS
4.1. The Riemann Integral	4.1 Notes	-	4.1 Video (14:22)	4.1 Transcript
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)	4.2 Transcript
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)	4.3 Transcript
Supplement: The Dirac Delta Function - A Cautionary Tale	Dirac Delta	-	Dirac Delta Video (24:08)	Dirac Delta Transcript
4.4. The General Lebesgue Integral	4.4 Notes	4.4 Supplement 4.4 Supplement Print File	4.4 Video (57:40)	4.4 Transcript
4.5. Countable Additivity and Continuity of Integration	4.5 Notes	4.5 Supplement 4.5 Supplement Print File	4.5 Video (10:43)	4.5 Transcript
4.6. Uniform Integrability: The Vitali Convergence Theorem	4.6 Notes	4.6 Supplement 4.6 Supplement Print File	4.6 Video (50:42)	4.6 Transcript

Chapter 5. Lebesgue Integration: Further Topics.

SECTION	NOTES	SUPPLEMENTS	VIDEOS	TRANSCRIPTS
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)	5.1 Transcript
5.2. Convergence in Measure	5.2 Notes	5.2 Supplement 5.2 Supplement Print File	5.2 Video (30:45)	5.2 Transcript
5.3. Characterization of Riemann and Lebesgue Integrability	5.3 Notes	5.3 Supplement 5.3 Supplement Print File	5.3 Video (32:12)	5.3 Transcript

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Last updated: November 28, 2020.