Measure Theory

This book intends to be a basic approach to measure theory.

In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the base set. Depending on the application, the "size" of a subset may be interpreted as (for example) its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of integration over domains with more complex structure than intervals of the real line. Such integrals are used extensively in probability theory, and in much of mathematical analysis.

It is often not possible or desirable to assign a size to all subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a σ-algebra

• 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.