Measure Theory

Set Theory/Systems of sets 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 Basic Structures And Definitions 00% Jul 12, 2006 Semialgebras, Algebras and σ-algebras 25% Jul 12, 2006 Measures 25% Dec 16, 2012 Measurable Functions 00% Dec 16, 2012 Extension of a Measure 00% Jul 12, 2006 Completion of Measure Spaces 00% Jul 12, 2006 Regular Measures 00% Jul 12, 2006 /Integration/ 50% Nov 3, 2008 /Riesz' representation theorem/ L^p spaces Contents /Advanced set theory/ /Algebras and σ-algebras/ /Pre-measures and measures/ /Theorems on measures/ /Multiplicative systems, Dynkin systems/ /Carathéodory's theorem and extension of pre-measures/ /Measurable functions, Lebesgue integration/ /Theorems on Lebesgue integrals/ (note to self: don't forget trafo of vars, leibniz integral rule) /Lp spaces/ /Riesz representation theorem, Radon–Nikodym theorem/ References Bartle, Robert Gardner. The Elements of Integration and Lebesgue Measure. Wiley, 1995. 192 p. 0471042226 DiBenedetto, Emmanuele. Real Analysis. Springer, 2002. 420 p. 0817642315 Folland, Gerald B.. Real Analysis: Modern Techniques and Their Applications. 2.ed. 1999. 408 p. 0471317160 Halmos, Paul Richard. Measure Theory. Springer, 1974. 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. 0024041513 W. Rudin, Real and Complex analysis, 3.ed. McGraw-Hill International (1987). 430 p. 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. Mathematical analysis University level mathematics books M ja:測度論 25% pt:Medida e integração