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Programmes → I.M.Sc. in Mathematical Sciences and M.Sc. in Statistics-OR

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Probability and Measure Theory

Course Code: ST 351; ST 452  |  Credits: 4  |  Prerequisite: Real Analysis-I, Elements of Probability and Statistics

Syllabus Details

  • Unit I:(Set Theory and Measures) Concepts from set theory, orderings, cardinality, well ordered sets, the axiom of choice, the extended real number system, metric spaces, the Bolzano-Weierstrass and Heine-Borel properties; σ-algebras, Borel σ-algebras, elementary family, measures, measurable space, measure space and properties of a measure; outer measures, Caratheodory’s theorem, pre-measure.
  • Unit II:(Integration) Borel measures on the real line, distribution function, Lebesgue-Stieltjes measure, Lebesgue measure, measurable functions, characteristic function; integration of non-negative functions, the monotone convergence theorem, Fatou’s lemma, integration of complex functions, the dominated convergence theorem.
  • Unit III:(Product Measures and Random Variables) Riemann integral, modes of convergence, Cauchy measure, product measures, the monotone class lemma, the Fubini-Tonelli theorem, signed measures, the Radon-Nikodym theorem (statement only); probability, distribution function, random variables, independence, functions of random variables, sequences of random variables and modes of convergence.
  • Unit IV:(Limit Theorems) Weak law of large numbers, Borel-Cantelli lemma, Kolmogorov’s inequality, strong law of large numbers, central limit theorem, construction of sample spaces.

Course Learning Objective

  1. Apply important concepts from set theory, well-ordered sets and metric spaces.
  2. Explain concepts such as outer measure, pre-measure and Caratheodory’s extension theorem.
  3. Explain the concepts of field, σ-field and measure on Borel σ-fields.
  4. Integrate non-negative and complex functions under measures.
  5. Explain different modes of convergence, independence of events and functions of random variables with examples.
  6. Apply the convergence of sequences of random variables in statistical analysis.
  7. Apply the weak and strong laws of large numbers and the central limit theorem.
  1. Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. 2nd ed. Wiley, 1999
  2. Ash, Robert B. and Doleans-Dade, Catherine A. Probability and Measure Theory. 2nd ed. Academic Press, 1999
  3. Billingsley, Patrick. Probability and Measure. Anniversary ed. Wiley, 2012
  4. Dudley, R. M. Real Analysis and Probability. Wadsworth and Brooks/Cole, 1989
  5. Kingman, J. F. C. and Taylor, S. J. Introduction to Measure and Probability. Cambridge University Press, 1966
  6. Bhat, B. R. Modern Probability Theory. 3rd ed. New Age International, 1999
  7. Capinski, Marek and Zastawniak, Tomasz. Probability Through Problems. Springer, 2001
  8. Chung, Kai Lai. A Course in Probability Theory. 2nd ed. Academic Press, 1974
  9. Feller, William. An Introduction to Probability Theory and its Applications, Volume 1. 3rd ed. Wiley, 1968
  10. Parzen, Emanuel. Modern Probability Theory and its Applications. Wiley Eastern, 1960
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