Section
Measure Theory
Sigma-algebras, measures, and foundations of Lebesgue integration.
Core idea
Sigma-Algebras
- Set algebra (algebra of sets)
- σ-algebra
- Borel σ-algebra
- Measurable space
- Measurable set
- Measurable rectangle
Measurable Functions
Measures
Definitions
Constructions
Properties
Lebesgue Measure
Lebesgue Integration
Basic Definitions
- Almost-everywhere equality
- Lebesgue integral (nonnegative functions)
- Lebesgue integral
- Lebesgue integrable function
L^p Spaces
Convergence
Product Measures
Convergence Theorems
- Monotone Convergence Theorem
- Fatou's Lemma
- Dominated Convergence Theorem
- Tonelli's Theorem
- Fubini's Theorem
Inequalities
Uncategorized
- Characteristic function (indicator function)
- Lebesgue number lemma refinement lemma
- Set of measure zero in ℝ^k
Integration and function-space additions
- Bochner integral
- Completeness of Lebesgue spaces
- Differentiation under an integral sign
- Hausdorff measure in a metric space
- Integrable majorant for a family
- Locally integrable function
- Interpolation between two Lebesgue norms
- Mixed Lebesgue norm
- Parabolic Hausdorff measure
- Sigma-finite measure
- Complete measure
- Sigma-algebra generated by a family
- Hausdorff dimension
- Completion of a measure space
- Product sigma-algebra
- Strongly measurable Banach-valued function