Książki

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 modelling, including the Black-Scholes formula, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework. In addition, further exercises and examples are provided to encourage the reader to become directly involved with the material.

Measure, Integral and Probability

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.