Measure Theory (Hardcover)

0. Conventions and Notation.- 1. Notation: Euclidean space.- 2. Operations involving ±?.- 3. Inequalities and inclusions.- 4. A space and its subsets.- 5. Notation: generation of classes of sets.- 6. Product sets.- 7. Dot notation for an index set.- 8. Notation: sets defined by conditions on functions.- 9. Notation: open and closed sets.- 10. Limit of a function at a point.- 11. Metric spaces.- 12. Standard metric space theorems.- 13. Pseudometric spaces.- I. Operations on Sets.- 1. Unions and intersections.- 2. The symmetric difference operator ?.- 3. Limit operations on set sequences.- 4. Probabilistic interpretation of sets and operations on them.- II. Classes of Subsets of a Space.- 1. Set algebras.- 2. Examples.- 3. The generation of set algebras.- 4. The Borel sets of a metric space.- 5. Products of set algebras.- 6. Monotone classes of sets.- III. Set Functions.- 1. Set function definitions.- 2. Extension of a finitely additive set function.- 3. Products of set functions.- 4. Heuristics on a algebras and integration.- 5. Measures and integrals on a countable space.- 6. Independence and conditional probability (preliminary discussion).- 7. Dependence examples.- 8. Inferior and superior limits of sequences of measurable sets.- 9. Mathematical counterparts of coin tossing.- 10. Setwise convergence of measure sequences.- 11. Outer measure.- 12. Outer measures of countable subsets of R.- 13. Distance on a set algebra defined by a subadditive set function.- 14. The pseudometric space defined by an outer measure.- 15. Nonadditive set functions.- IV. Measure Spaces.- 1. Completion of a measure space (S, S,?).- 2. Generalization of length on R.- 3. A general extension problem.- 4. Extension of a measure defined on a set algebra.- 5. Application to Borel measures.- 6. Strengthening of Theorem 5 when the metric space S is complete and separable.- 7. Continuity properties of monotone functions.- 8. The correspondence between monotone increasing functions on R and measures on B(R).- 9. Discrete and continuous distributions on R.- 10. Lebesgue-Stieltjes measures on RN and their corresponding monotone functions.- 11. Product measures.- 12. Examples of measures on RN.- 13. Marginal measures.- 14. Coin tossing.- 15. The Caratheodory measurability criterion.- 16. Measure hulls.- V. Measurable Functions.- 1. Function measurability.- 2. Function measurability properties.- 3. Measurability and sequential convergence.- 4. Baire functions.- 5. Joint distributions.- 6. Measures on function (coordinate) space.- 7. Applications of coordinate space measures.- 8. Mutually independent random variables on a probability space.- 9. Application of independence: the 0-1 law.- 10. Applications of the 0-1 law.- 11. A pseudometric for real valued measurable functions on a measure space.- 12. Convergence in measure.- 13. Convergence in measure vs. almost everywhere convergence.- 14. Almost everywhere convergence vs. uniform convergence.- 15. Function measurability vs. continuity.- 16. Measurable functions as approximated by continuous functions.- 17. Essential supremum and infimum of a measurable function.- 18. Essential supremum and infimum of a collection of measurable functions.- VI. Integration.- 1. The integral of a positive step function on a measure space (S, S,?,).- 2. The integral of a positive function.- 3. Integration to the limit for monotone increasing sequences of positive functions.- 4. Final definition of the integral.- 5. An elementary application of integration.- 6. Set functions defined by integrals.- 7. Uniform integrability test functions.- 8. Integration to the limit for positive integrands.- 9. The dominated convergence theorem.- 10. Integration over product measures.- 11. Jensen’s inequality.- 12. Conjugate spaces and Holder’s inequality.- 13. Minkowski’s inequality.- 14. The LP spaces as normed linear spaces.- 15. Approximation of LP functions.- 16. Uniform integrability.- 17. Uniform integrability in terms of uniform integrability test functions.- 18. L1 convergence and uniform integrability.- 19. The coordinate space context.- 20. The Riemann integral.- 21. Measure theory vs. premeasure theory analysis.- VII. Hilbert Space.- 1. Analysis of L2.- 2. Hilbert space.- 3. The distance from a subspace.- 4. Projections.- 5. Bounded linear functionals on h.- 6. Fourier series.- 7. Fourier series properties.- 8. Orthogonalization (Erhardt Schmidt procedure).- 9. Fourier trigonometric series.- 10. Two trigonometric integrals.- 11. Heuristic approach to the Fourier transform via Fourier series.- 12. The Fourier-Plancherel theorem.- 13. Ergodic theorems.- VIII. Convergence of Measure Sequences.- 1. Definition of convergence of a measure sequence.- 2. Linear functionals on subsets of ?(S).- 3. Generation of positive linear functionals by measures (S compact metric)..- 4. ?(5) convergence of sequences in M(S) (S compact metric).- 5. Metrization of M(s) to match ?(s) convergence; compactness of Mc(S) (S compact metric).- 6. Properties of the function ???[f], from M(S), in the dM metric into R (S compact metric).- 7. Generation of positive linear functionals on ?0(S) by measures (S a locally compact but not compact separable metric space).- 8. ?0(S)and?00(S)convergence of sequences in M(s) (S a locally compact but not compact separable metric space).- 9. Metrization of M(s) to match ?0(S) convergence; compactness of Mc(S) (S a locally compact but not compact separable metric space, c a strictly positive number).- 10. Properties of the function ???[f], from M(S) in the d0M metric into R (S a locally compact but not compact separable metric space).- 11. Stable?0(S) convergence of sequences in M (S) (S a locally compact but not compact separable metric space).- 12. Metrization of M(s) to match stable ?0(S) convergence (S a locally compact but not compact separable metric space).- 13. Properties of the function ???[f], from M(S) in the dM metric into R (S a locally compact but not compact separable metric space).- 14. Application to analytic and harmonic functions.- IX. Signed Measures.- 1. Range of values of a signed measure.- 2. Positive and negative components of a signed measure.- 3. Lattice property of the class of signed measures.- 4. Absolute continuity and singularity of a signed measure.- 5. Decomposition of a signed measure relative to a measure.- 6. A basic preparatory result on singularity.- 7. Integral representation of an absolutely continuous measure.- 8. Bounded linear functionals on L1.- 9. Sequences of signed measures.- 10. Vitali-Hahn-Saks theorem (continued).- 11. Theorem 10 for signed measures.- X. Measures and Functions of Bounded Variation on R.- 1. Introduction.- 2. Covering lemma.- 3. Vitali covering of a set.- 4. Derivation of Lebesgue-Stieltjes measures and of monotone functions.- 5. Functions of bounded variation.- 6. Functions of bounded variation vs. signed measures.- 7. Absolute continuity and singularity of a function of bounded variation.- 8. The convergence set of a sequence of monotone functions.- 9. Helly’s compactness theorem for sequences of monotone functions.- 10. Intervals of uniform convergence of a convergent sequence of monotone functions.- 11. ?(I) convergence of measure sequences on a compact interval I.- 12. ?0(R) convergence of a measure sequence.- 13. Stable ?0(R) convergence of a measure sequence.- 14. The characteristic function of a measure.- 15. Stable ?0(R) convergence of a sequence of probability distributions.- 16. Application to a stable ?0(R) metrization of M(R).- 17. General approach to derivation.- 18. A ratio limit lemma.- 19. Application to the boundary limits of harmonic functions.- XI. Conditional Expectations ; Martingale Theory.- 1. Stochastic processes.- 2. Conditional probability and expectation.- 3 Conditional expectation properties.- 4. Filtrations and adapted families of functions.- 5. Martingale theory definitions.- 6. Martingale examples.- 7. Elementary properties of (sub- super-) martingales.- 8. Optional times.- 9. Optional time properties.- 10. The optional sampling theorem.- 11. The maximal submartingale inequality.- 12. Upcrossings and convergence.- 13. The submartingale upcrossing inequality.- 14. Forward (sub- super-) martingale convergence.- 15. Backward martingale convergence.- 16. Backward supermartingale convergence.- 17. Application of martingale theory to derivation.- 18. Application of martingale theory to the 0-1 law.- 19. Application of martingale theory to the strong law of large numbers.- 20. Application of martingale theory to the convergence of infinite series.- 21. Application of martingale theory to the boundary limits of harmonic functions.- Notation.