• Spaces of measurable functions
• Convergence theorems
• Morphisms and categories of measure spaces
• The Caratheodory construction
• Measures on topological spaces
• Measures on Polish spaces
• Integral operators

Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals.

Chapters

1. Basic Structures And Definitions
   1. Semialgebras, Algebras and σ-algebras
   2. Measures
   3. Measurable Functions
   4. Extension of a Measure
   5. Completion of Measure Spaces
   6. Regular Measures
2. Integration
3. Riesz' representation theorem
4. L^p spaces

Contents

1. Advanced set theory
2. Algebras and σ-algebras
3. Pre-measures and measures
4. Theorems on measures
5. Multiplicative systems, Dynkin systems
6. Carathéodory's theorem and extension of pre-measures
7. Measurable functions, Lebesgue integration
8. Theorems on Lebesgue integrals (note to self: don't forget trafo of vars, leibniz integral rule)
9. Lp spaces
10. Riesz representation theorem, Radon–Nikodym theorem

References

• Bartle, Robert Gardner. The Elements of Integration and Lebesgue Measure. Wiley, 1995. 192 p. ISBN 0471042226
• DiBenedetto, Emmanuele. Real Analysis. Springer, 2002. 420 p. ISBN 0817642315
• Folland, Gerald B.. Real Analysis: Modern Techniques and Their Applications. 2.ed. 1999. 408 p. ISBN 0471317160
• Halmos, Paul Richard. Measure Theory. Springer, 1974. ISBN 0387900888
• Munroe, Marshall Evans. Introduction to Measure and Integration. 2.ed. Addison-Wesley, 1959. 310 p.
• Royden, M.. Real Analysis. New York: Collier Macmillan, 1988. ISBN 0024041513
• W. Rudin, Real and Complex analysis, 3.ed. McGraw-Hill International (1987). 430 p. ISBN 0070542341
• Lang, Serge. Analysis I. 3.ed. Addison-Wesley, 1973. 460 p.
• Tao, Terence: An introduction to measure theory

Users Actively Contributing

• Bunder :
  • At the moment, before making any writing of material i'm creating a sort of workflow by using templates
• Raliaga :
  • I'm adding content to the first section of the first chapter, including some insights about the "reason of being" of the definitions and comments on the importance of the propositions presented.
• SPat ^talk :
  • Basically uploading my lecture notes.