Week-1:Introduction to the course: A review of basic Probability and motivation towards the Mathematical formulation of Probability Theory; Fields and Sigma-fields of subsets of a non-empty set; Examples (emphasis on Borel sigma-fields on Euclidean spaces); Limits of sequences of events/sets, Monotone Class Theorem (Statement only)
Week-2:Measures and Measure spaces (Emphasis on Probabililty measures and Probability spaces); Examples and Properties
Week-3:Measurable functions (Emphasis on Random Variables); Examples:Properties (composition, algebraic properties, pointwise limits, measurability of components in higher dimensions); More examples using the above properties
Week-4:Caratheodery Extension Theorem (Emphasis on Uniqueness part, Existence part is statement only): Law/distribution of Random Variables; Distribution functions of Random Variables, properties
Week-5:Correspondence between Distribution functions and Probability measures on the real line; Extension of the above correspondence to higher dimensions (in brief); Lebesgue measure on the real line; Lebesgue measure on higher dimensions (in brief)
Week-6:Integration of measurable functions with respect to a measure (Emphasis on Expectation and Moments of Random Variables); Law/distribution of discrete Random variables, Example of measure theoretic integration: Expectation of discrete Random Variables; Convergence Theorems (Monotone Convergence Theorem, Fatou’s Lemma, Dominated Convergence Theorem), applications
Week-7:Convergence Theorems (Monotone Convergence Theorem, Fatou’s Lemma, Dominated Convergence Theorem), applications - Continued; Connection between Riemann and Lebesgue integration
Week-8:Radon-Nikodym Theorem (Statement only); Interpretation of the probabililty density function for absolutely continuous Random Variables as a Radom-Nikodym density, Law/distribution of absolutely continuous Random variables; Example of measure theoretic integration: Expectation of absolutely continuous Random Variables; inequalities involving Expectation and Moments of Random Variables; Conclusion of the course
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