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Measure Theory

Lebesgue measure and integral, σ-algebras, convergence theorems.

Mësime

Advanced σ-algebras and measures Which sets can be measured, and the countable additivity that makes limits work. integrate 1/10 dx from 2 to 5 Advanced The Lebesgue integral and convergence theorems Integrating by slicing the range instead of the domain; monotone and dominated convergence. integrate e^(-x) dx from 0 to oo

The Riemann integral breaks on functions that jump too much. Measure theory rebuilds integration from "size of a set" upward, and the Lebesgue integral that results is the one probability theory and analysis actually use.

Mësime

Measuring sets

Measure (mathematics)Σ-algebraBorel setLebesgue measureOuter measureNull setAlmost everywhereCarathéodory's criterionNon-measurable setVitali set

The integral

Lebesgue integrationMeasurable functionSimple functionMonotone convergence theoremDominated convergence theoremFatou's lemmaFubini's theoremRadon–Nikodym theoremLp spaceAbsolute continuity

Uses

Probability spaceExpected valueConvergence of random variablesSigned measureProduct measure

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What is wrong with the Riemann integral?

It fails on functions that oscillate too much, and it does not interact well with limits: the limit of integrable functions need not be integrable. Lebesgue's integral fixes both.

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