Measure and Integration (Real Analysis II)
Math 501
Spring 2006

Week 2 : Cartesian Product, Relations, Equivalence Relations, Orders, Chains, Partial and Linear Orders, Axiom of Choice, Zorn's Lemma, Hausdorff Maximal Principle, Well-Ordering Principle, Countable and Uncountable Sets, Cardinality, Equivalence of Cardinality, Theorem of Cantor, Cardinality of Sets of Real Numbers, Cantor Sets, Cantor Function

Week 3 : Concepts of Semi-Ring, Sigma-Sets, Ring, Sigma-Ring, Delta-Sets, Borel Sets of R^d, Algebra, Sigma-Algebra, Borel Sigma-Algebra of R^d, Dynkin Systems, Intersection-Stability, Smallest Sigma-Algebra Generated by Family of Sets, Introduction to Metric and Topological Spaces, Open Balls, Open Sets, Interior Points, Closed Sets, Closure, Closed Balls, Accumulation Points, Simple Convergence

Week 4 : Boundary Points, Boundary, Products of Metric Spaces, Dense Sets, Nowhere Dense Sets, Simple Continuity, Characterization of Continuous Mappings on Metric Spaces, Homeomorphism, Equivalent Metrics, Bounded Sets, Diameter of Sets, Cauchy Sequence, Completeness, Complete Metric Spaces, Space of Uniformly Bounded Functions B(Omega), Characterization of Closed Sets by Complete Metric SubSpaces, Theorem of Cantor, Separability, Examples of L^p and C^0, Completeness of L^p and C^0, Meager Sets, Baire Spaces, Baire's Category Theorem

Week 5 : Uniform Continuity, Continuity of Distances, Isometry, Completion, Isometric Metric Spaces, Contractive Mappings, Banach's Contraction Mapping Principle, Defect Inequality, Cover and Subcover, Theorem of Lindeloef, Compactness, Compact Metric Spaces, Characterization of Compact Sets, Proof of Theorem of Heine-Borel

Week 6 : Continuous Functions and Compactness, Total Boundedness, Isolated Points, Concepts of Topological Spaces, Topology, Open Sets, Discrete Topology, Induced Topology, Borel Sets of Topological Spaces, Neighborhood, Interior and Closure Points, Dense Sets, Boundary, Accumulation Points, Meager Sets, Nowhere Dense Sets, Hausdorff Space, Characterization of Continuous Mappings, Compact Topological Spaces, Oscillation Measure of Functions, Duality Relation of F_sigma and G_delta Sets, Compact Subsets

Week 7 : Extreme Value Behavior on Compact Sets, Homeomorphic Topological Spaces, Homeomorphism, Linear Vector and Normed Spaces, Norm, Linear Span, Basis, Linear Dependence, Completeness of l_p, Equivalence of Norms on Finite-dimensional Vector Spaces, Open and Closed Unit Balls, Local Compactness, Symmetric and Convex and Circled Sets, Vector Lattices, Function Spaces, The Spaces B(X) and C^0(X), Pointwise and Uniform Convergence of Functions on Topological Spaces, Completeness of C^0(X), Monotonicity, Dini's Theorem, Weierstrass and Dirichlet Tests, Arzela-Ascoli Theorem

Week 8 : Proof of Arzela-Ascoli Theorem, Topological Set Functions, Abstract Measures, Measures on Semirings, Properties of Abstract Measures: sigma-Monotonicity, sigma-Additivity, Positive and Monotone Measures, Subadditivity, Measurable Spaces, Measure Space, Examples: Counting Measure, Dirac Measure, Atomic Measure, f-Induced Measures, n-Dimensional Lebesque-Measure, Finite and sigma-Finite Measures, Contents, Additive and Finitely Additive Measures, Subtractivity of Positive Measures, Monotone Continuity of Measures (m-Continuity)

Week 9 : Spring Break

Week 10 : Equivalence of sigma-Additivity and m-Continuity, Minimal Extension-Ring, Trace of a Set, Lebesgue's Extension of Measures on Minimal Extension-Ring of a Semiring, Caratheodory's Extension of Measures by Outer Measures, Inner and Outer Positive Measures, Measurability of Sets, Properties of Outer Measures, Set of Measurable Sets M(X) over X, Complete Measure Spaces, Completion Procedure of Kolmogorov, Null Sets, Equivalence of Completion of Metric Spaces and Measure Spaces, Caratheodories Equivalence Theorem of Measurability, sigma-Subadditivity of Outer Measures

Week 11 : Examples of Null Sets, sigma Algebra Property of M(X), sigma-Additivity of Positive Measure Space (X,M(X),mu), Extension of n-dimensional Lebesgue Measure lambda_n on M(X), Unique Extension Properties of Outer Measures, Examples of Nonmeasurable Sets, Translation and Rotation Invariance of Lebesgue Measure lambda, Measurable Covers, Metric Outer Measure, Measurability of Closed and Open Sets, Signed Measures nu, Generation by sigma-Additive Measures, Generation by Signed Lebesgue-Stieltjes Measures, Properties of nu-Positive and nu-Negative Sets, Hahn Decomposition

Week 12 : Variation and Total Variation of Signed Measures, Jordan Decomposition, Properties of Signed Measures, The Space S(X,F), Vector Lattice Property of S(X,F), Linear Combination of Measures, Infimum and Supremum of Measures, The Complete Metric Space ((X,F,nu),rho), Singular and Absolute-Continuous Measures, Support of a Signed Measure, Properties and Relations between Absolute-Continuous Measures, Lebesgue Decomposition of Signed Measures, Radon-Nikodym Theorem, Borel Sets of and Borel Measures on Topological Spaces, Regular Borel Measure, Lebesgue Measure is a Regular Borel Measure, Measurable Functions and Mappings, Examples, Nonmeasurable Functions

Week 13 : Properties of Measurable Functions, Algebra M of Measurable Functions, Vector Lattice and Function Space Property of M, Convergence Almost Everywhere, Theorem of Egorov, Theorem of Luzin, mu-Equivalence of Functions, Convergence in Measure, Continuity of Lattice Operations on M, Theorem of Riesz, On Relationship Pointwise Convergence and Convergence in Measure, mu-Cauchy Sequences, Equivalence of Convergence Concepts under Monotonicity

Week 14 : Simple Functions, Integration of Simple Functions, Daniell's Properties of Simple Integrals, Lebesgue(-Stieltjes) Integral, Absolute-Continuity and Sigma-Additivity of Lebesgue Integral, Chebyshev Inequality, Markov Inequality, Convergence Theory for Lebesgue Integral, Proof of Bounded (Dominated) Convergence Theorem

Week 15 : Proofs of Lemma of Fatou, Levi's Theorem, Monotone Convergence Theorem, Application to Convergence of Function Series, Uniform Integrability, Main Convergence Theorem, Semicontinuous Functions, Baire's Limit Functions, Relation between Riemann and Lebesgue Integrals, Monotone Functions, Lebesgues Theorem on Differentiability of Monotone Functions, Functions of Bounded Variation, Total Variation, Subadditivity of Variations, Additivity of Limits, Further Properties

Week 16 : Banach Space of Functions of Bounded Variation, L^p Spaces, Embedding of L^p Spaces, Inequalities (Young, Hoelder, Cauchy-Bunjakowski-Schwarz, Minkowski, Fundamental, Jensen, Lyapunov), Product Measures, Additivity and Sigma-Additivity of Product Measures, Theorem of Fubini (Theorem of Fubini-Tonelli).

Further Interesting Related Topics / Miscallenea: Banach Space of Functions of Bounded Variation, Bounded p-Variation, Complete Proof of Radon Nikodym Theorem (Daniell Integral, Probability Measures, PseudoMeasures, Radon Measures, Weak Convergence, Tightness, Separations, Uryson's Lemma, Theorem of Uryson, Tietze's Extension Theorem, Stone-Weierstrass Theorem, Helley's Selection Principle, Hilbert Spaces and Operators)

Read this Without Tears: WHAT IS EXPECTED OF YOU (From "Teaching at the University Level" by Stephen Zucker, Notices Amer. Math. Soc. 43 (1996), p. 863))

25 Further Introductory Readings To Real Analysis, Measure and Integration -Theory For The BEYOND-THE-HORIZON Student (optional, for those they want to have profound readings I recommend to work through this list, and make your personal preferences, but do not expect to understand them all immediately):

40 Further Introductory Readings To Measure-Theory-Based Probability and Stochastic Processes For The BEYOND-THE-HORIZON Student (optional, for those they want to have profound readings in measure-axiomatic-based probability theory, I recommend to work through this list, and make your personal preferences, but do not expect to understand them all immediately, it took me more than 18 years, and we are still working on it!):