研究生一年级Analysis计划中

Measure theory and integration

σ-algebras and measures, Lebesgue measure, measurable functions and the Lebesgue integral, the monotone and dominated convergence theorems, product measures and Fubini–Tonelli, and Lᵖ spaces.

math.analysis.measure-theory

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前置知识

Real analysis Countability and Cantor's diagonal argument Metric spaces and point-set topology

38 项必需的前置知识（一直追溯到底）

你将学会

Construct Lebesgue measure from outer measure using Carathéodory's criterion.
Prove and apply the monotone and dominated convergence theorems.
Apply the Fubini–Tonelli theorem, and prove that Lᵖ spaces are complete.

定义

A σ-algebra on X is a collection of subsets of X that contains X and is closed under complements and countable unions.
A measure is a countably additive function μ from a σ-algebra to [0, ∞] with μ(∅) = 0.
Lᵖ(μ) consists of measurable functions f with ∫ |f|ᵖ dμ < ∞, with functions equal almost everywhere identified.

定理

Dominated convergence theorem.
Fubini–Tonelli theorem for σ-finite measure spaces.
Lᵖ(μ) is a Banach space for 1 ≤ p ≤ ∞ (Riesz–Fischer).
Every Riemann-integrable function on [a, b] is Lebesgue integrable, with the same integral.

常见误解

Thinking sets of measure zero must be countable; the Cantor set is uncountable and has measure zero.
Interchanging limits and integrals without a domination or monotonicity hypothesis.