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Lecture Notes for Integration
Aim of this lecture note is to develop an understanding of the statements of the theorems and how to apply them carefully. Major topics covered are: Measure spaces, Outer measure, null set, measurable set, The Cantor set, Lebesgue measure on the real line, Counting measure, Probability measures, Construction of a non-measurable set , Measurable function, simple function, integrable function, Reconciliation with the integral introduced in Prelims, Simple comparison theorem, Theorems of Fubini and Tonelli.
Author(s): Z. Qian and Alison Etheridge
NA Pages 
This graduate-level lecture note covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.
Author(s): Prof. Jeff Viaclovsky
NA Pages
This note covers the following topics: Elementary Integrals, Substitution, Trigonometric integrals, Integration by parts, Trigonometric substitutions, Partial Fractions.
Author(s): Mark MacLean and Andrew Rechnitzer
96 Pages
Integration Theory Lecture notes
This note introduces the concepts of measures, measurable functions and Lebesgue integrals. Topics covered includes: Measurable functions / random variables , Dynkin’s Lemma and the Uniqueness Theorem, Borel-Cantelli’s First Lemma, Independent random variables, Kolmogorov’s 0-1-law, Integration of nonnegative functions , Jordan-Hahn Decompositions, The Lebesgue-Radon-Nikodym Theorem, The law of large numbers.
Author(s): Johan Jonasson
71 Pages
This lecture note explains the following topics: The integral: properties and construction, Function spaces, Probability, Random walk and martingales, Radon integrals.
Author(s): William G. Faris
72 Pages