Measure Theory

Measure Theory and Probability Theory Summary: This post consolidates fundamental definitions and concepts into a single article.

Measure Defined on the Real Numbers $\mathbb{R}$

• Null Sets
• Outer Measure $m^\ast$
• Sigma Algebra and Measurable Space $(X, \mathcal{E})$
• Lebesgue Measure $m$
• Borel Sets
• Lebesgue Measurable Functions
• Almost Everywhere and Almost Surely in Measure Theory
• Expressing Any Function as Two Non-Negative Functions
  • Expressing the Absolute Value of Any Function as Two Non-Negative Functions

Integration

• Partition and Refinement of Measurable Spaces
• Lebesgue Integration
  • Lebesgue Integration as a Generalization of Riemann Integration
• Fatou’s Lemma
• Monotone Convergence Theorem
• Lebesgue Integrability
• Dominated Convergence Theorem
• Levi’s Theorem in Measure Theory

General Measures

• Algebras, Pre-Measures
• General Definition of Measure
  • Sigma-Finite Measures
  • Regular Measures
• Measurable Functions
  • Properties of Real-Valued Measurable Functions
  • Existence of Simple Function Sequences Converging to Measurable Functions
  • Egorov’s Theorem: Pointwise Converging Measurable Functions Converge Almost Uniformly
  • Lusin’s Theorem: Existence of Approximating Continuous Functions for Measurable Functions
• Borel $\sigma$-Algebras, Borel Measurable Spaces
• Necessary and Sufficient Condition for an Extended Real-Valued Function to be Measurable
• Carathéodory’s Theorem
• Absolute Continuity of Measures
  • Absolute Continuity of Real Functions
• Radon-Nikodym Derivative
• Radon-Nikodym Theorem
• Uniform Integrability
• Convergence of Measures
• Vitali Convergence Theorem
• Pi Systems and Lambda Systems
• Dynkin’s Pi-Lambda Theorem
• Weak Convergence of Measures

Signed Measures and Differentiation

• Signed Measures
• Positive Sets, Negative Sets, Null Sets
• Hahn Decomposition Theorem
• Mutually Singular
• Jordan Decomposition Theorem
• Total Variation
• Absolute Continuity of Signed Measures
• Lebesgue-Radon-Nikodym Auxiliary Theorem
• Relationship Between Absolutely Continuous and Integrable Functions
• Complex Measures, Vector Measures
• Maximal Inequality
• Hardy-Littlewood Maximal Function
• Maximal Theorem
• The Average Value of Locally Integrable Functions Converges to the Function Value at the Center

References

• Robert G. Bartle, The Elements of Integration and Lebesgue Measure (1995)
• Capinski, Measure, Integral and Probability (1999)
• Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (2nd Edition, 1999)