maths.free › Measure Theory

Measure Theory

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

सबक

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.

और पाठ

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

इनमें से कुछ सबक खुले पाठ से तैयार किए गए हैं: Wikipedia. हर पाठ पृष्ठ पर उसका स्रोत और लाइसेंस बताया गया है।

यहाँ उपयोग में संकेत

पूरी परिभाषा, चित्र, और इसमें हर अक्षर का अर्थ क्या है?

लोग जो सवाल पूछते हैं

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.

अपनी समस्या है?
सॉल्वर पेज पर इसे टाइप करें या बनाएं: हर चरण, एक चित्र, और उत्तर तब तक छिपा रहता है जब तक आप न पूछें। हल करें →

अन्य शाखाएँ

＋ 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