Search for tag: "further topics in analysis"
Measure Theory (14/15)  Lebesgue outer measure and Lebesgue measure on the real line (2/2)The sigmaalgebra 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…
From Joel Feinstein
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Measure Theory (13/15)  Lebesgue outer measure and Lebesgue measure on the real line (1/2)Brief discussion of the following: what we hope to achieve when measuring length, area and volume; the BanachTarski paradox; the Carathéodory extension theorem. The definition of Lebesgue…
From Joel Feinstein
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Measure Theory (12/15)  Properties of measures (2/2)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,…
From Joel Feinstein
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Measure Theory (11/15)  Properties of measures (1/2)Deduction of standard properties of measures from the two axioms: finite additivity; monotonicity; countable subadditivity. Continuity properties of measures stated and discussed. For more…
From Joel Feinstein
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Measure Theory (10/15)  Measurable spaces, measurable sets, measures and measure spaces (2/2)Terminology: countable additivity, pairwise disjoint unions. Notation for pairwise disjoint unions. Examples and constructions of measures: counting measure; combinations of measures; the biggest…
From Joel Feinstein
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Measure Theory (09/15)  Measurable spaces, measurable sets, measures and measure spaces (1/2)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 (nonnegative) measure on a…
From Joel Feinstein
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Measure Theory (08/15)  The sigmaalgebra generated by a collection of subsets (2/2)Properties of the sigmaalgebra on a set X generated by a collection of subsets. Methods for showing inclusion or equality of sigmaalgebras generated by two different collections of subsets. The…
From Joel Feinstein
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Measure Theory (05/15)  Collections of sets (1/2)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, …
From Joel Feinstein
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Measure Theory (04/15)  The algebra of limitsThe algebra of limits for sums of sequences of nonnegative extended real numbers. The algebra of limits for products, and its limitations, for sequences of nonnegative extended real numbers. For…
From Joel Feinstein
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Measure Theory (03/15)  Arithmetic and series in the nonnegative extended real numbersAddition and multiplication for nonnegative extended real numbers. Problems with the cancellation laws. Series of nonnegative extended real numbers. Some discussion of problems with the algebra of…
From Joel Feinstein
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Measure Theory (02/15)  The extended real line (2/2)Discussion of convergence to infinity for sequences of nonnegative extended real numbers. Monotone sequences (nondecreasing or nonincreasing) and the Monotone Sequence Theorem for the extended real…
From Joel Feinstein
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Measure Theory (01/15)  The extended real line (1/2)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…
From Joel Feinstein
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Measure Theory (07/15)  The sigmaalgebra generated by a collection of subsets (1/2)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 subbase), and…
From Joel Feinstein
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Measure Theory (06/15)  Collections of sets (2/2)Audio narration (with digital pointer) to go with the annotations of slides 3536 from G14FTA Further Topics in Analysis, where the original recording failed. More examples of σalgebras:…
From Joel Feinstein
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