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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…
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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…
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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,…
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Deduction of standard properties of measures from the two axioms: finite additivity; monotonicity; countable subadditivity. Continuity properties of measures stated and discussed. For
more…
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Terminology: countable additivity, pairwise disjoint unions. Notation for pairwise disjoint unions. Examples and constructions of measures: counting measure; combinations of measures; the biggest…
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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…
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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…
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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, …
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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…
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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…
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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…
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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…
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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…
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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:…
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