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 24 subcategories, out of 24 total.
A
D
F
I
L
M
N
O
P
S
T
Pages in category "Measure Theory"
The following 179 pages are in this category, out of 179 total.
A
B
C
- Cantor Set has Zero Lebesgue Measure
- Carathéodory's Theorem (Measure Theory)
- Characteristic Function Measurable iff Set Measurable
- Characterization of Essentially Bounded Functions
- Characterization of Integrable 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
- Convergence a.u. Implies Convergence a.e.
- Convergence in Mean implies Convergence in Measure
- 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
- Functions A.E. Equal iff Positive and Negative Parts A.E. Equal
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 with Zero Integral on Sub-Sigma-Algebra is A.E. Zero
- 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 Measurable Set is Well-Defined
- 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 as Limit of Integrals of Positive Simple Functions
- 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 is 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 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/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 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 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
P
- Piecewise Combination of Measurable Mappings is Measurable/Binary Case
- Piecewise Combination of Measurable Mappings is Measurable/General Case
- Pointwise Convergence implies Convergence Almost Everywhere
- Pointwise Convergence Implies Convergence in Measure
- Positive Part of Multiple of 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