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This book offers a straightforward introduction to the mathematical theory of probability. It presents the central results and techniques of the subject in a complete and self-contained account. As a result, the emphasis is on giving results in simple forms with clear proofs and to eschew more powerful forms of theorems which require technically involved proofs. Throughout there are a wide variety of exercises to illustrate and to develop ideas in the text.

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´Prelude: Random Walks.- The Model.- Random variables.- Probability.- First calculations.- Issues and Approaches.- Issues.- Approaches.- Tools.- Functional of the Random Walk.- Times of returns to the origin.- Numbers of returns to the origin.- First passage times.- Maxima.- Time spent positive.- Limit Theorems.- Summary.- 1 Probability.- 1.1 Random Experiments and Sample Spaces.- 1.1.1 Random experiments.- 1.1.2 Sample spaces.- 1.2 Events and Classes of Sets.- 1.2.1 Events.- 1.2.2 Basic set operations.- 1.2.3 Indicator functions.- 1.2.4 Operations on sequences of sets.- 1.2.5 Classes of sets closed under set operations.- 1.2.6 Generated classes.- 1.2.7 The monotone class theorem.- 1.2.8 Events, bis.- 1.3 Probabilities and Probability Spaces.- 1.3.1 Probability.- 1.3.2 Elementary properties.- 1.3.3 More advanced properties.- 1.3.4 Almost sure and null events.- 1.3.5 Uniqueness.- 1.4 Probabilities on R.- 1.4.1 Distribution functions.- 1.4.2 Discrete probabilities.- 1.4.3 Absolutely continuous probabilities.- 1.4.4 Mixed distributions.- 1.5 Conditional Probability Given a Set.- 1.6 Complements.- 1.6.1 The extended real numbers.- 1.6.2 Measures.- 1.6.3 Lebesgue measure.- 1.6.4 Singular probabilities on R.- 1.6.5 Representation of probabilities on R.- 1.7 Exercises.- 2 Random Variables.- 2.1 Fundamentals.- 2.1.1 Random variables.- 2.1.2 Random vectors.- 2.1.3 Stochastic processes.- 2.1.4 Complex-valued random variables.- 2.1.5 The -algebra generated by a random variable.- 2.1.6 Simplified criteria.- 2.2 Combining Random Variables.- 2.2.1 Algebraic operations.- 2.2.2 Limiting operations.- 2.2.3 Transformations.- 2.2.4 Approximation of positive random variables.- 2.2.5 Monotone class theorems.- 2.3 Distributions and Distribution Functions.- 2.3.1 Random variables.- 2.3.2 Random vectors.- 2.4 Key Random Variables and Distributions.- 2.4.1 Discrete random variables.- 2.4.2 Absolutely continuous random variables.- 2.4.3 Random vectors.- 2.5 Transformation Theory.- 2.5.1 Random variables.- 2.5.2 Random vectors.- 2.6 Random Variables with Prescribed Distributions.- 2.6.1 Individual random variables.- 2.6.2 Random vectors.- 2.6.3 Sequences of random variables.- 2.7 Complements.- 2.7.1 Measurability with respect to sub-?-algebras.- 2.7.2 Borel measurable functions.- 2.8 Exercises.- 3 Independence.- 3.1 Independent Random Variables.- 3.1.1 Fundamentals.- 3.1.2 Criteria for independence.- 3.1.3 Examples.- 3.2 Functions of Independent Random Variables.- 3.2.1 Transformation properties.- 3.2.2 Sums of independent random variables.- 3.3 Constructing Independent Random Variables.- 3.3.1 Finite families.- 3.3.2 Sequences.- 3.4 Independent Events.- 3.5 Occupancy Models.- 3.5.1 Four occupancy models.- 3.5.2 Occupancy numbers.- 3.5.3 Asymptotics.- 3.6 Bernoulli and Poisson Processes.- 3.6.1 Bernoulli processes.- 3.6.2 Poisson processes.- 3.7 Complements.- 3.7.1 Independent ?-algebras.- 3.7.2 Products of probability spaces.- 3.8 Exercises.- 4 Expectation.- 4.1 Definition and Fundamental Properties.- 4.1.1 Simple random variables.- 4.1.2 Positive random variables.- 4.1.3 Integrable random variables.- 4.1.4 Complex-valued random variables.- 4.2 Integrals with respect to Distribution Functions.- 4.2.1 Generalities.- 4.2.2 Discrete distribution functions.- 4.2.3 Absolutely continuous distribution functions.- 4.2.4 Mixed distribution functions.- 4.3 Computation of Expectations.- 4.3.1 Positive random variables.- 4.3.2 Integrable random variables.- 4.3.3 Functions of random variables.- 4.3.4 Functions of random vectors.- 4.3.5 Functions of independent random variables.- 4.3.6 Sums of independent random variables.- 4.4 LP Spaces and Inequalities.- 4.4.1 LPspaces.- 4.4.2 Key inequalities.- 4.5 Moments.- 4.5.1 Moments of random variables.- 4.5.2 Variance and standard deviation.- 4.5.3 Covariance and correlation.- 4.5.4 Moments of random vectors.- 4.5.5 Multivari

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