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► Measure Theory Definitions

► Measure Theory Notes

► Measure Theory Theorems

► Measure Theory Theorems, lemmas and corollaries

  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

  Borel sigma-algebra

  Borel sigma-algebra generated by

  Borel sigma-algebra of the real line

  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-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

  Extended real value

  Extended real valued

  Extended real values

  Extending pre-measures to measures

  Extending pre-measures to outer-measures

  Finitely additive set function

  Function terminology

  Generator

  Generator (sigma-algebra)

  Hereditary

  Hereditary (measure theory)

  Hereditary class of sets

  Hereditary set system

  Notes:Hereditary sigma-ring

  Hereditary sigma-ring

  Hereditary sigma-ring generated by

  Hereditary system

  Hereditary system generated by

  Hereditary system of sets

  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

  Lebesgue measure

  Liminf (sequence of sets)

  Limit of increasing sequence of sets

  Limsup (sequence of sets)

  Limsup and liminf (sequence of sets)

  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

  Outer splicing set

  Outer-measure

  Outer-measure/Definition

  Notes:Outer-measures to measures

  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

  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

  Random variable

  Random variables

  Real-valued function

  Ring (measure theory)

  Ring generated by

  Ring of sets

  Ring of sets/Definition

  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

  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