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Measure Theory

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

ບົດຮຽນ

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.

ບົດຮຽນ​ເພີ່ມເຕີມ

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

ບົດຮຽນບາງບົດນີ້ ແມ່ນໄດ້ນຳເອົາຈາກບົດຂຽນເປີດ: Wikipedia. ທຸກໆ​ໜ້າ​ບົດຮຽນ​ມີ​ຊື່​ແຫຼ່ງ​ຂອງ​ມັນ ແລະ ໃບອະນຸຍາດ​ຂອງ​ມັນ.

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ກົດ​ຕົວ​ອັກສອນ​ໃດ​ໜຶ່ງ​ເພື່ອ​ເບິ່ງ​ຄວາມ​ໝາຍ​ເຕັມ, ຮູບ ແລະ ຕົວອັກສອນ​ແຕ່ລະ​ຕົວ​ໃນ​ມັນ​ໝາຍ​ຄວາມ​ວ່າ​ແນວໃດ.

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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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