# 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 15 subcategories, out of 15 total.

### D

### F

### I

### L

### M

### N

### O

### P

### S

## Pages in category "Measure Theory"

The following 196 pages are in this category, out of 196 total.

### A

- A.E. Equal Positive Measurable Functions have Equal Integrals
- Absolute Value of Simple Function is Simple Function
- Absolute Value of Simple Function is Simple Function/Proof 1
- Absolute Value of Simple Function is Simple Function/Proof 2
- Additive and Countably Subadditive Function is Countably Additive

### B

### C

- Cantor Set has Zero Lebesgue Measure
- Carathéodory's Theorem (Measure Theory)
- Characteristic Function Measurable iff Set Measurable
- Characterization of Integrable Functions
- Characterization of Measurable Functions
- Characterization of Measures
- Characterization of Pre-Measures
- Co-Countable Measure is Measure
- Co-Countable Measure is Probability Measure
- Completion Theorem (Measure Spaces)
- Completion Theorem (Measure Spaces)/Lemma
- Composition of Measurable Mappings is Measurable
- Construction of Outer Measure
- Continuity under Integral Sign
- Continuous Composition of Measurable Functions into Second Countable Space is Measurable
- Continuous Mapping 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
- Convolution of Measures as Pushforward Measure
- Convolution of Measures is Bilinear
- Countably Additive Function also Finitely Additive
- Counting Measure is Measure
- Cover of Interval By Closed Intervals is not Pairwise Disjoint

### D

### 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
- Fubini's Theorem
- Function Measurable iff Positive and Negative Parts Measurable

### I

- Induced Outer Measure Restricted to Semiring is Pre-Measure
- Infinite Measure is Measure
- 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 under Pushforward Measure
- Integrable Function Zero A.E. iff Absolute Value has Zero Integral
- Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal
- Integral of Characteristic Function
- Integral of Characteristic Function/Corollary
- Integral of Distribution Function
- Integral of Increasing Function Composed with Measurable Function
- Integral of Increasing Function Composed with Measurable Function/Corollary
- Integral of Integrable Function is Additive
- Integral of Integrable Function is Homogeneous
- Integral of Integrable Function is Monotone
- Integral of Integrable Function over Null Set
- Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions
- Integral of Positive Measurable Function Extends Integral of Positive Simple Function
- Integral of Positive Measurable Function is Additive
- Integral of Positive Measurable Function is Monotone
- Integral of Positive Measurable Function is Positive Homogeneous
- Integral of Positive Simple Function is Additive
- Integral of Positive Simple Function is Increasing
- Integral of Positive Simple Function is Positive Homogeneous
- Integral of Positive Simple Function Well-Defined
- Integral of Series of Positive Measurable Functions
- Integral of Survival Function
- Integral with respect to Dirac Measure
- Integral with respect to Discrete Measure
- Integral with respect to Kernel Transformation of Measure
- Integral with respect to Pushforward Measure
- Integral with respect to Series of Measures
- Intersection Measure is Measure

### J

### K

### L

- Lebesgue Integral is Extension of Darboux Integral
- Lebesgue Measure Invariant under Orthogonal Group
- Lebesgue Measure is Diffuse
- Lebesgue Measure is Translation-Invariant
- Lebesgue Measure of Matrix Image
- Lebesgue Measure of Scalar Multiple
- Lebesgue Pre-Measure is Pre-Measure
- Lebesgue's Dominated Convergence Theorem
- Linear Combination of Measures

### M

- Mapping between Euclidean Spaces Measurable iff Components Measurable
- Mapping Measurable iff Measurable on Generator
- Markov's Inequality
- Measurable Function is Simple Function iff Finite Image Set
- Measurable Function is Simple Function iff Finite Image Set/Corollary
- Measurable Function Pointwise Limit of Simple Functions
- Measurable Functions Determine Measurable Sets
- 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
- Measurable Sets form Algebra of Sets
- Measurable Sets form Sigma-Algebra
- Measure Invariant on Generator is Invariant
- Measure is Countably Subadditive
- Measure is Finitely Additive Function
- Measure is Monotone
- Measure is Strongly Additive
- Measure is Subadditive
- Measure is Subadditive/Corollary
- Measure of Empty Set is Zero
- Measure of Interval is Length
- Measure of Set Difference with Subset
- Measure of Stieltjes Function of Measure
- Measure Space from Outer Measure
- Measure Space Sigma-Finite iff Cover by Sets of Finite Measure
- Measure with Density is Measure
- Minkowski's Inequality for Double Integrals
- Monotone Convergence Theorem (Measure Theory)
- Monotone Convergence Theorem (Measure Theory)/Corollary

### N

### O

### P

- Piecewise Combination of Measurable Mappings is Measurable
- Piecewise Combination of Measurable Mappings is Measurable/Binary Case
- Piecewise Combination of Measurable Mappings is Measurable/General Case
- Pointwise Convergence Implies Convergence in Measure
- Pointwise Difference of Measurable Functions is Measurable
- Pointwise Infimum of Measurable Functions is Measurable
- Pointwise Limit of Measurable Functions is Measurable
- Pointwise Lower Limit of Measurable Functions is Measurable
- Pointwise Maximum of Measurable Functions is Measurable
- Pointwise Minimum of Measurable Functions is Measurable
- Pointwise Product of Measurable Functions is Measurable
- Pointwise Product of Simple Functions is Simple Function
- Pointwise Sum of Measurable Functions is Measurable
- Pointwise Sum of Simple Functions is Simple Function
- Pointwise Supremum of Measurable Functions is Measurable
- Pointwise Upper Limit of Measurable Functions is Measurable
- Positive Part of Simple Function is Simple Function
- Pratt's Lemma
- Pre-Image Sigma-Algebra on Domain is Generated by Mapping
- Pre-Measure of Finite Stieltjes Function Extends to Unique Measure
- Pre-Measure of Finite Stieltjes Function is Pre-Measure
- Probability Measure is Subadditive
- Pushforward Measure is Measure
- Pushforward of Lebesgue Measure under General Linear Group

### R

### S

- Series of Measures is Measure
- Series of Positive Measurable Functions is Positive Measurable Function
- Sigma-Algebras with Independent Generators are Independent
- Simple Function has Standard Representation
- Simple Function is Measurable
- Standard Machinery
- Stieltjes Function of Measure is Stieltjes Function
- Stieltjes Function of Measure of Finite Stieltjes Function
- Sum of Integrals on Complementary Sets