Measure Theory

Core idea

Sigma-Algebras

• Set algebra (algebra of sets)
• σ-algebra
• Borel σ-algebra
• Measurable space
• Measurable set
• Measurable rectangle

Measurable Functions

• Measurable function
• Indicator function
• Simple function

Measures

Definitions

• Premeasure
• Measure
• Measure space
• Outer measure
• Null set
• Almost everywhere

Constructions

• Carathéodory measurable set
• Carathéodory construction

Properties

• Continuity from above
• Continuity from below

Lebesgue Measure

• Jordan content
• Lebesgue measure
• Lebesgue criterion for Riemann integrability

Lebesgue Integration

Basic Definitions

• Almost-everywhere equality
• Lebesgue integral (nonnegative functions)
• Lebesgue integral
• Lebesgue integrable function

L^p Spaces

• Essential supremum
• L^p norm
• L^p space
• L¹ function
• L^∞ function

Convergence

• Convergence almost everywhere
• Convergence in measure
• Convergence in L^p
• Uniform integrability

Product Measures

• Product measure
• Pushforward measure
• Change of variables (pushforward)

Convergence Theorems

• Monotone Convergence Theorem
• Fatou's Lemma
• Dominated Convergence Theorem
• Tonelli's Theorem
• Fubini's Theorem

Inequalities

• Jensen's Inequality (integral form)
• Minkowski Inequality (L^p)

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

Evolution-equation prerequisites

• Ordered integration simplex

Normalized means

• Zero-mean function
• Projection onto zero-mean functions

Cones, averages and moment constraints

• Averaged second-moment matrix

Heat flow and integral profiles

• Reference-subtracted integral

Connections with ergodic theory

• Standard Borel space
• Isomorphism of probability spaces modulo null sets
• Standard probability space
• Atom of a measure
• Atomless probability space
• Isomorphism theorem for atomless standard probability spaces
• Hilbert space of square-integrable observables
• Probability kernel
• Disintegration of a probability measure

See the ergodic theory reading paths for the surrounding constructions and theorems.