**Aims & Learning Objectives:**
**Aims: **The purpose of this course is to lay the basic technical foundations and establish the main principles which underpin the classical notions of area, volume and the related idea of an integral.
**Objectives: **
The objective is to familiarise students with measure as a tool in analysis, functional analysis and probability theory. Students will be able to quote and apply the main inequalities in the subject, and to understand their significance in a wide range of contexts. Students will obtain a full understanding of the Lebesgue Integral.
**Content: ** Topics will be chosen from the following:
Measurability for sets: algebras, σ-algebras, π-systems, d-systems; Dynkin's Lemma; Borel σ-algebras. Measure in the abstract: additive and s-additive set functions; monotone-convergence properties; Uniqueness Lemma; statement of Caratheodory's Theorem and discussion of the l-set concept used in its proof; full proof on handout. Lebesgue measure on 3^{n}: existence; inner and outer regularity. Measurable functions. Sums, products, composition, lim sups, etc; The Monotone-Class Theorem. Probability. Sample space, events, random variables. Independence; rigorous statement of the Strong Law for coin tossing. Integration. Integral of a non-negative functions as sup of the integrals of simple non-negative functions dominated by it. Monotone-Convergence Theorem; 'Additivity'; Fatou's Lemma; integral of 'signed' function; definition of* L*^{p} and of *L*^{p} linearity; Dominated-Convergence Theorem - with mention that it is not the `right' result. Product measures: definition; uniqueness; existence; Fubini's Theorem. Absolutely continuous measures: the idea; effect on integrals. Statement of the Radon-Nikodým Theorem. Inequalities: Jensen, Holder, Minkowski. Completeness of *L*^{p}.
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