First-year graduateAnalysisPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 38
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Leads to
Optional discoveries
Learning pathways
Further reading
- Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications
- Elias M. Stein and Rami Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces (2005)