Measure, Integral and Probability

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. Content -- 1. Motivation And Preliminaries -- 1.1 Notation And Basic Set Theory -- 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 -- 2.7 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 -- 3.6 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 -- 4.8 Proofs Of Propositions -- 5. Spaces Of Integrable Functions -- 5.1 The Space L1 -- 5.2 The Hilbert Space L2 -- 5.3 The Lp Spaces: Completeness -- 5.4 Probability -- 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 -- 6.6 Proofs Of Propositions -- 7. The Radon—nikodym Theorem -- 7.1 Densities And Conditioning -- 7.2 The Radon—nikodym Theorem -- 7.3 Lebesgue—stieltjes Measures -- 7.4 Probability -- 7.5 Proofs Of Propositions -- 8. Limitl Theorems -- 8.1 Modes Of Convergence -- 8.2 Probability -- 8.3 Proofs Of Propositions -- Solutions -- References. By Marek Capiński, Peter Ekkehard Kopp. The key concept is that of measure which is first developed on the real line and then presented abstractly to provide an introduction to the foundations of probability theory (the Kolmogorov axioms) which in turn opens a route to many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities. Throughout, the development of the Lebesgue Integral provides the essential ideas: the role of basic convergence theorems, a discussion of modes of convergence for measurable functions, relations to the Riemann integral and the fundamental theorem of calculus, leading to the definition of Lebesgue spaces, the Fubini and Radon-Nikodym Theorems and their roles in describing the properties of random variables and their distributions. Applications to probability include laws of large numbers and the central limit theorem.