Integration and Probability (Hardcover, 1995)

I Measurable Spaces and Integrable Functions.- 1 ?-algebras.- 1.1 Sub-?-algebras. Intersection of ?-algebras.- 1.2 ?-algebra generated by a family of sets.- 1.3 Limit of a monotone sequence of sets.- 1.4 Theorem (Boolean algebras and monotone classes).- 1.5 Product ?-algebras.- 2 Measurable Spaces.- 2.1 Inverse image of a ?-algebra.- 2.2 Closure under inverse images of the generated ?-algebra.- 2.3 Measurable spaces and measurable mappings.- 2.4 Borel algebras. Measurability and continuity. Operations on measurable functions.- 2.5 Pointwise convergence of measurable mappings.- 2.6 Supremum of a sequence of measurable functions.- 3 Measures and Measure Spaces.- 3.1 Convexity inequality.- 3.2 Measure of limits of monotone sequences.- 3.3 Countable convexity inequality.- 4 Negligible Sets and Classes of Measurable Mappings.- 4.1 Negligible sets.- 4.2 Complete measure spaces.- 4.3 The space Mμ((X, A); (X?,A?)).- 5 Convergence in Mμ ((X,A);(Y,BY)).- 5.1 Convergence almost everywhere.- 5.2 Convergence in measure.- 6 The Space of Integrable Functions.- 6.1 Simple measurable functions.- 6.2 Finite ?-algebras.- 6.3 Simple functions and indicator functions.- 6.4 Approximation by simple functions.- 6.5 Integrable simple functions.- 6.6 Some spaces of bounded measurable functions.- 6.7 The truncation operator.- 6.8 Construction of L1.- 7 Theorems on Passage to the Limit under the Integral Sign.- 7.1 Fatou-Beppo Levi theorem.- 7.2 Lebesgue's theorem on series.- 7.3 Theorem (truncation operator a contraction).- 7.4 Integrability criteria.- 7.5 Definition of the integral on a measurable set.- 7.6 Lebesgue's dominated convergence theorem.- 7.7 Fatou's lemma.- 7.8 Applications of the dominated convergence theorem to integrals which depend on a parameter.- 8 Product Measures and the Fubini-Lebesgue Theorem.- 8.1 Definition of the product measure.- 8.2 Proposition (uniqueness of the product measure).- 8.3 Lemma (measurability of sections).- 8.4 Construction of the product measure.- 8.5 The Fubini-Lebesgue theorem.- 9 The Lp Spaces.- 9.0 Integration of complex-valued functions.- 9.1 Definition of the Lp spaces.- 9.2 Convexity inequalities.- 9.3 Completeness theorem.- 9.4 Notions of duality.- 9.5 The space L?.- 9.6 Theorem (containment relations between Lp spaces if μ(X) ?).- II Borel Measures and Radon Measures.- 1 Locally Compact Spaces and Partitions of Unity.- 1.0 Definition of locally compact spaces which are countable at infinity.- 1.1 Urysohn's lemma.- 1.2 Support of a function.- 1.3 Subordinate covers.- 1.4 Partitions of unity.- 2 Positive Linear Functionals onCK(X) and Positive Radon Measures.- 2.1 Borel measures.- 2.2 Radon-Riesz theorem.- 2.3 Proof of uniqueness of the Riesz representation.- 2.4 Proof of existence of the Riesz representation.- 3 Regularity of Borel Measures and Lusin's Theorem.- 3.1 Proposition (Borel measures and Radon measures).- 3.2 Theorem (regularity of Radon measures).- 3.3 Theorem (regularity of locally finite Borel measures).- 3.4 The classes G?(X) and F?(X).- 3.5 Theorem (density of CK in Lp).- 4 The Lebesgue Integral on R and on Rn.- 4.1 Definition of the Lebesgue integral on R.- 4.2 Properties of the Lebesgue integral.- 4.3 Lebesgue measure on Rn.- 4.4 Change of variables in the Lebesgue integral on Rn.- 5 Linear Functionals on CK(X) and Signed Radon Measures.- 5.1 Continuous linear functionals on C(X) (X compact).- 5.2 Decomposition theorem.- 5.3 Signed Borel measures.- 5.4 Dirac measures and discrete measures.- 5.5 Support of a signed Radon measure.- 6 Measures and Duality with Respect to Spaces of Continuous Functions on a Locally Compact Space.- 6.1 Definitions.- 6.2 Proposition (relationships among CbCK, and C0)..- 6.3 The Alexandroff compactification.- 6.4 Proposition.- 6.5 The space M1(X).- 6.6 Theorem (M1(X) the dual of C0(X)).- 6.7 Defining convergence by duality.- 6.8 Theorem (relationships among types of convergence).- 6.9 Theorem (narrow density of Md,f1in M1).- III Fourier Analysis.- 1 Convo