maths.free › Measure Theory

Measure Theory

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

Nodarbības

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.

Vairāk mācību

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

Dažas no šīm mācībām ir pielāgotas, ņemot vērā atklātos tekstus: Wikipedia. Katras nodarbības lapas nosaukums tās avots un licence.

Šeit izmantotie simboli

Piesitiet visiem simboliem, kas apzīmē pilnu definīciju, attēlu un ko katrs burts tajā nozīmē.

Jautājumi, ko cilvēki vaicā

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.

Vai jums ir kāda problēma?
Ievadiet to vai zīmējiet to uz atrisinātāja lapas: katrs solis, attēls, atbilde slēpta, līdz jūs jautājat. Atrisināt →

Citas filiāles

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers