Measure, Integral and Probability is a gentle introduction that makes measure and integration theory accessible to the average third-year undergraduate student. The ideas are developed at an easy pace in a form that is suitable for self-study, with an emphasis on clear explanations and concrete examples rather than abstract theory. For this second edition, the text has been thoroughly revised and expanded. New features include: · a substantial new chapter, featuring a constructive proof of the Radon-Nikodym theorem, an analysis of the structure of Lebesgue-Stieltjes measures, the Hahn-Jordan decomposition, and a brief introduction to martingales · key aspects of financial modeling, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework.

Inhalt

Content.- 1. Motivation and preliminaries.- 1.1 Notation and basic set theory.- 1.1.1 Sets and functions.- 1.1.2 Countable and uncountable sets in ?.- 1.1.3 Topological properties of sets in ?.- 1.2 The Riemann integral: scope and limitations.- 1.3 Choosing numbers at random.- 2. Measure.- 2.1 Null sets.- 2.2 Outer measure.- 2.3 Lebesgue-measurable sets and Lebesgue measure.- 2.4 Basic properties of Lebesgue measure.- 2.5 Borel sets.- 2.6 Probability.- 2.6.1 Probability space.- 2.6.2 Events: conditioning and independence.- 2.6.3 Applications to mathematical finance.- 2.7 Proofs of propositions.- 3. Measurable functions.- 3.1 The extended real line.- 3.2 Lebesgue-measurable functions.- 3.3 Examples.- 3.4 Properties.- 3.5 Probability.- 3.5.1 Random variables.- 3.5.2 ?-fields generated by random variables.- 3.5.3 Probability distributions.- 3.5.4 Independence of random variables.- 3.5.5 Applications to mathematical finance Proofs of propositions.- 3.6 Proofs of propositions.- 4. Integral.- 4.1 Definition of the integral.- 4.2 Monotone convergence theorems.- 4.3 Integrable functions.- 4.4 The dominated convergence theorem.- 4.5 Relation to the Riemann integral.- 4.6 Approximation of measurable functions.- 4.7 Probability.- 4.7.1 Integration with respect to probability distributions.- 4.7.2 Absolutely continuous measures: examples of densities.- 4.7.3 Expectation of a random variable.- 4.7.4 Characteristic function.- 4.7.5 Applications to mathematical finance.- 4.8 Proofs of propositions.- 5. Spaces of integrable functions.- 5.1 The space L1.- 5.2 The Hilbert space L2.- 5.2.1 Properties of the L2-norm.- 5.2.2 Inner product spaces.- 5.2.3 Orthogonality and projections.- 5.3 The LP spaces: completeness.- 5.4 Probability.- 5.4.1 Moments.- 5.4.2 Independence.- 5.4.3 Conditional expectation (first construction).- 5.5 Proofs of propositions.- 6. Product measures.- 6.1 Multi-dimensional Lebesgue measure.- 6.2 Product ?-fields.- 6.3 Construction of the product measure.- 6.4 Fubini's theorem.- 6.5 Probability.- 6.5.1 Joint distributions.- 6.5.2 Independence again.- 6.5.3 Conditional probability.- 6.5.4 Characteristic functions determine distributions.- 6.5.5 Application to mathematical finance.- 6.6 Proofs of propositions.- 7. The Radon-Nikodym theorem.- 7.1 Densities and conditioning.- 7.2 The Radon-Nikodym theorem.- 7.3 Lebesgue-Stieltjes measures.- 7.3.1 Construction of Lebesgue-Stieltjes measures.- 7.3.2 Absolute continuity of functions.- 7.3.3 Functions of bounded variation.- 7.3.4 Signed measures.- 7.3.5 Hahn-Jordan decomposition.- 7.4 Probability.- 7.4.1 Conditional expectation relative to a ?-field.- 7.4.2 Martingales.- 7.4.3 Doob decomposition.- 7.4.4 Applications to mathematical finance.- 7.5 Proofs of propositions.- 8. LimitL theorems.- 8.1 Modes of convergence.- 8.2 Probability.- 8.2.1 Convergence in probability.- 8.2.2 Weak law of large numbers.- 8.2.3 The Bore-Cantelli lemmas.- 8.2.4 Strong law of large numbers.- 8.2.5 Weak convergence.- 8.2.6 Central limit theorem.- 8.2.7 Applications to mathematical finance.- 8.3 Proofs of propositions.- Solutions.- References.