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### Topics Covered

These will be the chapters covered from the textbook:

Marek Capinski & Ekkehard Kopp, Measure, Integral and Probability, 2nd ed., Springer-Verlag, Springer Undergraduate Mathematics Series, ISBN 1-85233-781-8.

Chapter 1: Cantor's diagonal Argument, Riemann Integration.
Chapter 2: Null Sets, Cantor Set, Outer Measure (+ properties), Lebesgue Measure (+ properties), Sigma Fields, Borel Measure, Completions of Measure spaces, Probability Spaces.
Chapter 3: Measurable Functions, Properties of Measurable Functions (including examples, limits etc), Random Variables, Probability Distributions, Indpendence of Random Variables. We will not cover chapter 3.5.5 on mathematical finance).
Chapter 4: Lebesgue Integral of simple functions and of non-negative measurable functions, some basic properties of these integrals, Fatou's lemma (with examples), monotone convergence theorem (MCT) with examples and the proof of additivity of integral using MCT. Integrable functions along with some basic properties, dominated convergence theorem (DCT), examples where DCT holds and does not hold, Beppo-Levi theorem, use of DCT and Beppo-Levi to compute integrals. Comparison of Riemann and Lebesgue integral, approximation of measurable functions by continuous and step functions in the L1 sense and the Riemann-Lebesgue theorem. Probability distributions+ proof of uniqueness, cumulative distribution functions, expectation of a random variable, characteristic function. Again we will not cover chapter 4.7.5 on mathematical finance.
Chapter 5: Normed vector spaces, completeness, L1 (+ proof that it is a complete normed vector space), L2, Schwarz Inequality, Inner prodct spaces, Hilbert Spaces and proof that L2 is a Hilbert space. Parallelogram Law and orthogonal projections. Lp spaces, Hölder's inequality, Minkowski's inequality, proof that Lp is a normed vector space. Relationships between Lp and Lq for p not equal to q and examples. We will not cover Section 5.4 on probability.
Chapter 6: Instead of Sections 6.1-6.4, we covered material from here. Here we covered the definition of product measures, monotone classes, construction of the product measure, Fubini's theorem, examples where Fubini's theorem holds and also does not hold. We also explained how Fubini's theorem holds for completions of product measures. We also covered Section 6.5.1 on joint probability distributions and joint density functions. Section 6.5.2 on indpendence was covered and conditional probability in Section 6.5.3 was also covered. We did not cover conditional expectation from Section 6.5.3. We also did not cover Sections 6.5.4 and 6.5.5 and so these Sections will not be examined.
Chapter 7: Section 7.1 on densities and conditioning and Section 7.2 the Radon-Nikodym theorem and the Lebesgue decomposition theorem. You will not need to know the contents of chapters 7.3, 7.4, 7.5 and chapter 8.

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