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Measure Theory

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

Уроки

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.

Оқулықтар

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

Оның кейбір сабақтары ашық мәтіннен алынған: Wikipedia. Әрбір сабақ парағының өзінің көзі мен лицензиясы көрсетілген.

Мұнда қолданылатын символдар

Тұтас анықтамасын, суретін және әрбір әріптің мағынасын көру үшін кез келген символды түртіңіз.

Адамдар қоятын сұрақтар

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.

Өзіңіздің проблемаңыз бар ма?
Оны шешу парағының бетіне жазып не суреттеп қойыңыз: әрбір қадамы сурет, сұрағанша жауап жасырылады. & Шегіну →

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