Dive into a comprehensive 5-hour video series on measure theory, exploring fundamental concepts from sigma algebras to the Riemann and Lebesgue integrals. Learn about Borel sigma algebras, measurable maps, and key theorems including monotone convergence, Fatou's Lemma, and Lebesgue's dominated convergence. Examine Carathéodory's extension theorem, Lebesgue-Stieltjes measures, and the Radon-Nikodym theorem. Investigate image measures, the substitution rule, product measures, and Cavalieri's principle. Conclude with an in-depth look at outer measures and a comparison of Riemann and Lebesgue integrals, providing a solid foundation in this crucial area of mathematical analysis.
Overview
Syllabus
Measure Theory - Part 1 - Sigma algebra.
Measure Theory - Part 2 - Borel Sigma algebra.
Measure Theory - Part 3 - What is a measure?.
Measure Theory - Part 4 - Not everything is Lebesgue measurable.
Measure Theory - Part 5 - Measurable maps.
Measure Theory - Part 6 - Lebesgue integral.
Measure Theory - Part 7 - Monotone convergence theorem (and more).
Measure Theory - Part 8 - Monotone convergence theorem (Proof and application).
Measure Theory - Part 9 - Fatou's Lemma.
Measure Theory - Part 10 - Lebesgue's dominated convergence theorem.
Measure Theory - Part 11 - Proof of Lebesgue's dominated convergence theorem.
Carathéodory's extension theorem (Measure Theory Part 12).
Lebesgue-Stieltjes measures (Measure Theory Part 13).
Radon–Nikodym theorem and Lebesgue's decomposition theorem (Measure Theory Part 14).
Image measure and substitution rule (Measure Theory Part 15).
Proof of the substitution rule for measure spaces (Measure Theory Part 16).
Product measure and Cavalieri's principle (Measure Theory Part 17).
Cavalieri's principle - An example (Measure Theory Part 18).
Fubini's Theorem (Measure Theory Part 19).
Outer measures - Part 1 (Measure Theory Part 20).
Outer measures - Part 2: Examples (Measure Theory Part 21).
Outer measures - Part 3: Proof (Measure Theory Part 22).
Riemann integral vs. Lebesgue integral.
Taught by
The Bright Side of Mathematics
