Primer año de posgradoAnalysisPlanificado
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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Se basa en
Requisitos previos obligatorios, hasta el fondo: 38
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Conduce a
Descubrimientos opcionales
Itinerarios de aprendizaje
Lecturas complementarias
- 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)