Difference between revisions of "Category:Measure Theory"

A

• A collection of subsets is a sigma-algebra if and only if it is both a p-system and a d-system
• A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections
• A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)
• A map from two sigma-algebras, A and B, is measurable if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G
• A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
• Additive function
• Additive set function
• Algebra
• Algebra (disambiguation)
• Algebra (measure theory)
• Algebra of sets
• Algebra of sets/Infobox
• Algebras of sets

B

• Borel sigma-algebra
• Borel sigma-algebra generated by
• Borel sigma-algebra of the real line

C

• Class of sets closed under complements properties
• Class of smooth real-valued functions on R-n
• Class of smooth real-valued functions on R-n/Structure
• Classes of continuously differentiable functions
• Complementation
• Composition of measurable maps is measurable
• Conditions for a Dynkin system to be a sigma-algebra
• Conditions for a generated Dynkin system to be a sigma-algebra
• Conditions for a map to be a measurable map
• Countably additive set function

D

• D-system
• Derivative
• Dynkin system
• Dynkin system generated by
• Dynkin system/Definition 1
• Dynkin system/Definition 2
• Dynkin system/Proof that definitions 1 and 2 are equivalent
• Dynkin systems

E

• Extended real value
• Extended real valued
• Extended real values
• Extending pre-measures to measures
• Extending pre-measures to outer-measures

F

• Finitely additive set function
• Function terminology

G

• Generator
• Generator (sigma-algebra)

H

• Hereditary
• Hereditary (measure theory)
• Hereditary class of sets
• Hereditary set system
• Notes:Hereditary sigma-ring
• Hereditary sigma-ring
• Hereditary sigma-ring generated by

H cont.

• Hereditary system
• Hereditary system generated by
• Hereditary system of sets

I

• Index of common sigma-algebras
• Integral
• Integral (measure theory)
• Integral of a positive function (measure theory)
• Integral of a positive function (measure theory)/Definition
• Integral of a simple function (measure theory)
• Integral of a simple function (measure theory)/Definition

L

• Lebesgue measure
• Liminf (sequence of sets)
• Limit of increasing sequence of sets
• Limsup (sequence of sets)
• Limsup and liminf (sequence of sets)

M

• Measurable function
• Measurable functions
• Measurable map
• Measurable space
• Measure
• Measure space
• Measure Theory
• Measure Theory (subject)
• Template:Measure theory navbox
• Notes:Measure theory plan
• Measure theory terminology doctrine
• Notes:Measures
• Notes:Measures - Real and Abstract Analysis - Hewitt & Stromberg
• Measures are monotonic and subtractive
• Min/max
• Minimum
• Minimum function
• Monotone
• Monotone convergence theorem for non-negative numerical measurable functions
• Monotone convergence theorem for non-negative numerical measurable functions/Statement
• Monotonic
• Monotonic set function
• Monotonicity of the integral of non-negative extended-real-valued measurable functions with respect to a measure
• Mu*-measurable set

O

• Outer splicing set
• Outer-measure
• Outer-measure/Definition
• Notes:Outer-measures to measures

P

• P-system
• Positive and negative parts of a function
• Pre-image sigma-algebra
• Pre-image sigma-algebra/Definition
• Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra
• Pre-measurable space
• Pre-measure

P cont.

• Pre-measure on a semi-ring
• Pre-measure space
• Pre-measure/New page
• Pre-measure/Properties in common with measure
• Premeasurable space
• Premeasurable space/Definition
• Probability function
• Probability measure
• Probability space
• Properties of classes of sets closed under set-subtraction

R

• Random variable
• Random variables
• Real-valued function
• Ring (measure theory)
• Ring generated by
• Ring of sets
• Ring of sets/Definition

S

• Semi-ring (measure theory)
• Semi-ring of half-closed-half-open intervals
• Semi-ring of sets
• Semi-ring of sets/Definition
• Semiring (measure theory)
• Semiring of sets
• Set function
• Sigma-algebra
• Sigma-algebra generated by
• Sigma-algebra/Definition
• Sigma-algebras
• Sigma-field
• Sigma-ring
• Simple function (measure theory)
• Simple function (measure theory)/Definition
• Notes:Simple function approximation to a numerical function
• Simple function under-approximation to a numerical function
• Subtractive set function
• Symmetric difference

T

• The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set
• The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement
• The intersection of an arbitrary family of Dynkin systems is itself a Dynkin system
• The ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions
• The set of all mu*-measurable sets forms a ring
• The set of all mu*-measurable sets forms a sigma-ring
• The set of all mu*-measurable sets is a ring
• Notes:The set of all mu*-measurable sets is a ring
• The set of all mu*-measurable sets is a sigma-ring
• Trace sigma-algebra
• Trace sigma-algebra/Proof of claim that it actually is a sigma-algebra
• Trivial
• Types of set algebras