To introduce the concepts of sets, classes of events, measure and integral with respect to a measure, random variables and function and probability measure to show their basic properties, and to provide a basis for further studies in Analysis, Probability, and Dynamical Systems.

To describe algebra of sets, relations and the events and classes of events

To gain understanding of the abstract measure theory and definition and main properties of the integral

To construct Lebesgue’s measure on the real line and in n-dimensional Euclidean space

To explain the basic advanced directions of the theory

To gain the knowledge of probability measure, distribution functions and expectations, convergence of random variable and distribution

Having successfully completed this course, students will be able to demonstrate knowledge and understanding of:

State and prove theorems using measure theory

Use the definition of measurable functions to prove limit theorems

Define and use the Lebesgue integral

Use conditional expectations and martingales

Use product integration for calculating multivariate probability distribution functions

Sets and Classes of Events: Algebra of sets, Relations, Open and closed set on Rn, Events and classes of events.

Measure: s-Algebra, Measurable set, Measurability, Lebesgue measure on the real line, Properties of measures, Borel set.

Random Variable: Concept, Limit, Simple, Inverse, Measurable, Borel, Characteristic, Random variables as measurable functions.

Integral of Measurable Function: Lebesgue integral of simple, integrable, and sequences of integrable functions, General and Riemann-Stieltje’s integral.

Probability Measure: Concept, Simple properties, Discrete, general, and induced probability spaces, Extended probability, Probability measure, Lebesgue-Stieltje’s measure, Signed measure, Borel-Cantelli lemmas, Zero-one law, Kolmogorov’s zero-one law.

Distribution Function and Expectation: Distribution function (DF) of a random variable and a random vector, Decomposition of DF’s, Correspondence theorem, Expectation and its properties, moments and inequalities.

Convergence: Types– probability, almost sure, r-th mean, distribution and their relations, Convergence of distribution functions, characteristic functions, and moments.

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