# Category:Measure Theory

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This category contains results about **Measure Theory**.

Definitions specific to this category can be found in Definitions/Measure Theory.

**Measure theory** is the subfield of analysis concerned with the properties of measures, particularly the Lebesgue measure.

## Subcategories

This category has the following 51 subcategories, out of 51 total.

### A

### C

- Convergence in Mean (1 P)
- Convergence in Measure (2 P)
- Countably Additive Functions (2 P)

### D

- Dynkin Systems (8 P)

### F

- Fatou's Lemma (3 P)
- Fatou's Lemma for Integrals (3 P)
- Fatou's Lemma for Measures (4 P)
- Finitely Additive Functions (1 P)

### H

- Horizontal Section of Functions (12 P)

### I

### L

- Lebesgue Measure (10 P)

### M

- Markov's Inequality (2 P)

### N

- Negative Parts (13 P)

### O

### P

- Positive Parts (13 P)
- Pre-Measures (5 P)

### R

- Reverse Fatou's Lemma (3 P)

### S

- Scheffé's Lemma (2 P)
- Stieltjes Functions (2 P)

### T

### V

- Vertical Section of Functions (12 P)
- Vertical Section of Sets (11 P)
- Vitali Set Existence Theorem (4 P)
- Vitali Sets (2 P)

### Y

## Pages in category "Measure Theory"

The following 92 pages are in this category, out of 92 total.

### A

### C

- Carathéodory's Theorem (Measure Theory)
- Carathéodory's Theorem (Measure Theory)/Corollary
- Characteristic Function of Null Set is A.E. Equal to Zero
- Characterization of Almost Everywhere Zero
- Characterization of Essentially Bounded Functions
- Characterization of Integrable Functions
- Characterization of Measures
- Composition of Measurable Mappings is Measurable
- Continuity under Integral Sign
- Continuous Composition of Measurable Functions into Second Countable Space is Measurable
- Convergence a.u. Implies Convergence a.e.
- Convergence in Norm Implies Convergence in Measure
- Convergence in Sigma-Finite Measure
- Convolution of Integrable Function with Bounded Function
- Convolution of Measurable Function and Measure is Bilinear
- Convolution of Measurable Functions is Bilinear
- Countable Union of Measurable Sets as Disjoint Union of Measurable Sets
- Countably Additive Function also Finitely Additive
- Cover of Interval By Closed Intervals is not Pairwise Disjoint

### E

### F

- Factorization Lemma
- Factorization Lemma for Extended Real-Valued Functions
- Factorization Lemma for Real-Valued Functions
- Factorization Lemma/Extended Real-Valued Function
- Factorization Lemma/Real-Valued Function
- Fatou's Lemma for Integrals
- Fatou's Lemma for Integrals/Integrable Functions
- Fatou's Lemma for Integrals/Positive Measurable Functions
- Fatou's Lemma for Measures
- Finite Hausdorff Measure Implies Zero Higher Dimensional Measure
- Functions A.E. Equal iff Positive and Negative Parts A.E. Equal

### I

- Inner Limit in Hausdorff Space by Open Neighborhoods
- Inner Limit in Hausdorff Space by Set Closures
- Integrable Function is A.E. Real-Valued
- Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero
- Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero/Proof 2
- Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal
- Integral of Distribution Function

### J

### L

### M

- Mapping between Euclidean Spaces Measurable iff Components Measurable
- Mapping Measurable iff Measurable on Generator
- Markov's Inequality
- Measurable Function is Integrable iff A.E. Equal to Real-Valued Integrable Function
- Measurable Function is Simple Function iff Finite Image Set/Corollary
- Measurable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal
- Measurable Image
- Measurable Mappings from Product Measurable Space
- Measurable Mappings to Product Measurable Space
- Measure of Stieltjes Function of Measure
- Minkowski's Inequality for Double Integrals
- Monotone Convergence Theorem (Measure Theory)
- Monotone Convergence Theorem (Measure Theory)/Corollary
- Monotone Convergence Theorem for Positive Simple Functions

### S

- Scheffé's Lemma
- Series of Positive Measurable Functions is Positive Measurable Function
- Sigma-Algebras with Independent Generators are Independent
- Simple Function has Standard Representation
- Standard Machinery
- Stieltjes Function of Measure is Stieltjes Function
- Stieltjes Function of Measure of Finite Stieltjes Function
- Strictly Positive Hausdorff Measure implies Infinite Lower Dimensional Measure