Measure theory and integration

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 small in every sense; the Cantor set is uncountable and null, and there are null sets that are dense G_δ sets (topologically large).
• Interchanging limits and integrals without a domination or monotonicity hypothesis.

Conduce a

Descubrimientos opcionales

• ← Area, perimeter and angles
• ← Probability theory
• ← The axiom of choice and its equivalents
• ← Descriptive set theory

Itinerarios de aprendizaje

• Graduate preparation
• Analysis, PDE and dynamics
• Mathematical physics

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)