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Measure Theory

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

Poučení

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.

Další lekce

Measuring sets

Measure (mathematics)Σ-algebra Borel set Lebesgue measure Outer measure Null set Almost everywhere Carathéodory's criterion Non-measurable set Vitali set

The integral

Lebesgue integration Measurable function Simple function Monotone convergence theorem Dominated convergence theorem Fatou's lemma Fubini's theorem Radon–Nikodym theorem Lp space Absolute continuity

Uses

Probability space Expected value Convergence of random variables Signed measure Product measure

Některé z těchto lekcí jsou přizpůsobeny otevřeným textům: Wikipedia. Každá lekční stránka jmenuje svůj zdroj a licenci.

Zde použité symboly

Ťukněte na jakýkoliv symbol pro úplnou definici, obrázek a co každé písmeno v něm znamená.

Otázky, které se lidé ptají

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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