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
Integration
- Partition and Refinement of Measurable Spaces
- Lebesgue 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
- 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
- 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)
All posts
- Empty Set
- Exterior Measure
- Sigma Algebra and Measurable Spaces
- Lebesgue Measure
- Borel Set
- Lebesgue Measurable Functions
- In Measure Theory: Almost Everywhere and Almost Surely
- Lebesgue Integration
- Proof of Fatou's Lemma
- Monotone Convergence Theorem Proof
- Lebesgue Integrable
- Proof of the Dominated Convergence Theorem
- Proof of Levy's Theorem in Measure Theory
- Lebesgue Integral as a Generalization of Riemann Integral
- Proof of the Karatheodory's Theorem
- Predictable Functions
- Properties of Measurable Functions with Real Values
- Methods of Expressing an Arbitrary Function as Two Non-negative Functions
- Borel Sigma-Algebra, Borel Measurable Space
- Conditions for a Function with Extended Real Values to be Measurable
- General Definitions of Measure
- Signed Measures
- Positive Set, Negative Set, Null Set
- Hahn Decomposition Theorem
- Mutually Singular
- Jordan Decomposition Theorem
- Partition and Refinement of Measurable Spaces
- Absolute Continuity of Measures
- Finite Sigma Measures
- Radon-Nikodym Derivative
- Radon-Nikodym Theorem Proof
- Total Variation
- Absolute Continuity of the Sign Measure
- Lebesgue-Radon-Nikodym Theorem
- Relationship between Absolutely Continuous and Integrable Functions
- Methods for Expressing the Absolute Value of an Arbitrary Function as Two Non-negative Functions
- Algebra, Quasi-measure
- Complex Measures, Vector Measures
- Hardy-Littlewood Maximal Function
- Maximal Lemma
- Maximal Theorem
- The Mean Value of Locally Integrable Functions Converges to the Value of the Function at the Center.
- Uniform Integrability
- Convergence of Measures
- Vitali Convergence Theorem
- Pi System and Lambda System
- Dinkin's Pi-Lambda Theorem
- Convergence in Measure
- Regular Measure
- Yegorov's Theorem
- Lusin's Theorem
- Existence of a Sequence of Simple Functions Converging to a Measurable Function
- Absolutely Continuous Real Function