Course

Measure theory and integration

Course under construction · 5 of 32 concepts published; more are added as they are reviewed.

You learn σ-algebras, Lebesgue measure and integral, their convergence theorems, Lp spaces, and the Fubini-Tonelli and Radon-Nikodym theorems, tools for rigorously formalizing analysis and probability. It helps to have taken Discrete Mathematics, Real Analysis and Set Theory.

Recommended before this course: Discrete mathematics · Real analysis · Set theory (2 concepts come from those courses and are tagged)

7concepts

4topics

70exercises

≈ 4 hof study approx.

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Course path

1
Lebesgue measure 2 concepts
Vitali set · Axiom of choice from Set theory
2
Lebesgue integral 1 concepts
Measurable function
3
Lp spaces 3 concepts
Minkowski inequality · Riesz–Fischer theorem · Cauchy sequence from Real analysis
4
Product and signed measures 1 concepts
Hahn–Jordan decomposition

Sources

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, 1999.
H. L. Royden, P. M. Fitzpatrick, Real Analysis, 4th edition, Pearson, 2010.
Herbert B. Enderton, Elements of Set Theory, 1st edition, Academic Press, 1977.
Thomas Jech, Set Theory (The Third Millennium Edition), 3rd millennium edition, Springer, 2003.
Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015.
Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011.