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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 (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

Dynkin system generated by

Dynkin system/Definition 1

Dynkin system/Definition 2

Dynkin system/Proof that definitions 1 and 2 are equivalent

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 (sigma-algebra)

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 (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 space

Measure Theory (subject)

Template:Measure theory navbox

Notes:Measure theory plan

Measure theory terminology doctrine

Notes:Measures - Real and Abstract Analysis - Hewitt & Stromberg

Measures are monotonic and subtractive

Minimum function

Monotone convergence theorem for non-negative numerical measurable functions

Monotone convergence theorem for non-negative numerical measurable functions/Statement

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/Definition

Notes:Outer-measures to measures

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 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/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

Sigma-algebra generated by

Sigma-algebra/Definition

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

Types of set algebras

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