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The sigma-algebra of Lebesgue measurable sets. Lebesgue measure λ (on the Lebesgue measurable sets or on the Borel sets). Further properties of Lebesgue measure λ and Lebesgue outer…

Brief discussion of the following: what we hope to achieve when measuring length, area and volume; the Banach-Tarski paradox; the Carathéodory extension theorem. The definition of Lebesgue…

Proofs of the continuity properties of measures for nested increasing unions and nested decreasing intersections of sequences of measurable sets. For
more information and links to further resources,…

Deduction of standard properties of measures from the two axioms: finite additivity; monotonicity; countable subadditivity. Continuity properties of measures stated and discussed. For
more…

Terminology: countable additivity, pairwise disjoint unions. Notation for pairwise disjoint unions. Examples and constructions of measures: counting measure; combinations of measures; the biggest…

Measurable spaces and measurable sets. Brief discussion of length, area and volume, the idea behind Lebesgue measure, and some of the issues. The definition of a (non-negative) measure on a…

Properties of the sigma-algebra on a set X generated by a collection of subsets. Methods for showing inclusion or equality of sigma-algebras generated by two different collections of subsets. The…

Sets of sets, which we call collections of sets.Comparison of the properties of some collections of sets, including topologies, and the collection of closed sets in a topological space.Definition, …

The algebra of limits for sums of sequences of non-negative extended real numbers. The algebra of limits for products, and its limitations, for sequences of non-negative extended real numbers. For…

Addition and multiplication for non-negative extended real numbers. Problems with the cancellation laws. Series of non-negative extended real numbers. Some discussion of problems with the algebra of…

Discussion of convergence to infinity for sequences of non-negative extended real numbers. Monotone sequences (nondecreasing or nonincreasing) and the Monotone Sequence Theorem for the extended real…

An introduction to the extended real line, with its total order (extending the total order on the real line), and a metric (induced, via a suitable bijection, by the usual metric on [-1,1]). Brief…

Audio narration (with digital pointer) to go with Slide 37 of G14FTA where the original recording failed. Brief discussion of how to generate a topology from a collection of sets (as a sub-base), and…

Audio narration (with digital pointer) to go with the annotations of slides 35-36 from G14FTA Further Topics in Analysis, where the original recording failed. More examples of σ-algebras:…