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 12 subcategories, out of 12 total.
D
I
L
N
O
P
S
Pages in category "Measure Theory"
The following 194 pages are in this category, out of 194 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 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
- Fatou's Lemma for Measures/Corollary
- 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 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 Riemann 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 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
