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Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsigaclcuni 24501* A sigma-algebra is closed under countable union: indexed union version (Contributed by Thierry Arnoux, 8-Jun-2017.)
sigAlgebra

Theoremsigaclfu 24502 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra

Theoremsigaclcu2 24503* A sigma-algebra is closed under countable union - indexing on (Contributed by Thierry Arnoux, 29-Dec-2016.)
sigAlgebra

Theoremsigaclfu2 24504* A sigma-algebra is closed under finite union - indexing on ..^ (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra ..^ ..^

Theoremsigaclcu3 24505* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
sigAlgebra       ..^

Theoremissgon 24506 Property of being a sigma-algebra with a given base set, noting that the base set of a sigma algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
sigAlgebra sigAlgebra

Theoremsgon 24507 A sigma alebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra sigAlgebra

Theoremelsigass 24508 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremelrnsiga 24509 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
sigAlgebra sigAlgebra

Theoremisrnsigau 24510* The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
sigAlgebra

Theoremunielsiga 24511 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremdmvlsiga 24512 Lebesgue-measurable subsets of form a sigma-algebra (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
sigAlgebra

Theorempwsiga 24513 Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
sigAlgebra

Theoremprsiga 24514 The smallest possible sigma-algebra containing (Contributed by Thierry Arnoux, 13-Sep-2016.)
sigAlgebra

Theoremsigaclci 24515 A sigma-algebra is closed under countable intersection. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
sigAlgebra

Theoremdifelsiga 24516 A sigma algebra is closed under set difference. (Contributed by Thierry Arnoux, 13-Sep-2016.)
sigAlgebra

Theoremunelsiga 24517 A sigma algebra is closed under set union. (Contributed by Thierry Arnoux, 13-Dec-2016.)
sigAlgebra

Theoreminelsiga 24518 A sigma algebra is closed under set intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.)
sigAlgebra

Theoremsigainb 24519 Building a sigma algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
sigAlgebra sigAlgebra

Theoreminsiga 24520 The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigAlgebra sigAlgebra

19.3.13.2  Generated Sigma-Algebra

Syntaxcsigagen 24521 Extend class notation to include the sigma-algebra generator.
sigaGen

Definitiondf-sigagen 24522* Define the sigma algebra generated by a given collection of sets as the intersection of all sigma algebra containing that set. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen sigAlgebra

Theoremsigagenval 24523* Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen sigAlgebra

Theoremsigagensiga 24524 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen sigAlgebra

Theoremsgsiga 24525 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
sigaGen sigAlgebra

Theoremunisg 24526 The sigma algebra generated by a collection is a sigma algebra on . (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen

Theoremdmsigagen 24527 A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
sigaGen

Theoremsssigagen 24528 A set is a subset of the sigma algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
sigaGen

Theoremsssigagen2 24529 A subset of the generating set is also a subset of the generated sigma algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
sigaGen

Theoremelsigagen 24530 Any element of set is also an element of the sigma algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
sigaGen

Theoremelsigagen2 24531 Any countable union of elements of a set is also in the sigma algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
sigaGen

Theoremsigagenss 24532 The generated sigma-algebra is a subset of all sigma algebra containing the generating set, i.e. the generated sigma-algebra is the smallest sigma algebra containing the generating set, here . (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra sigaGen

Theoremsigagenss2 24533 Sufficient condition for inclusion of sigma algebra. This is used to prove equality of sigma algebra. (Contributed by Thierry Arnoux, 10-Oct-2017.)
sigaGen sigaGen sigaGen

Theoremsigagenid 24534 The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra sigaGen

19.3.13.3  The Borel algebra on the real numbers

Syntaxcbrsiga 24535 The Borel Algebra on real numbers, usually a gothic B
𝔅

Definitiondf-brsiga 24536 A Borel Algebra is defined as a sigma algebra generated by a topology. 'The' Borel sigma algebra here refers to the sigma algebra generated by the topology of open intervals on real numbers. The Borel algebra of a given topology is the sigma-algebra generated by , sigaGen, so there is no need to introduce a special constant function for Borel sigma Algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigaGen

Theorembrsiga 24537 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigaGen

Theorembrsigarn 24538 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigAlgebra

Theorembrsigasspwrn 24539 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
𝔅

Theoremunibrsiga 24540 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝔅

Theoremcldssbrsiga 24541 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
sigaGen

19.3.13.4  Product Sigma-Algebra

Syntaxcsx 24542 Extend class notation with the product sigma-algebra operation.
×s

Definitiondf-sx 24543* Define the product sigma-algebra operation, analogue to df-tx 17594. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s sigaGen

Theoremsxval 24544* Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s sigaGen

Theoremsxsiga 24545 A product sigma-algebra is a sigma-algebra (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s sigAlgebra

Theoremsxsigon 24546 A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s sigAlgebra

Theoremsxuni 24547 The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s

Theoremelsx 24548 The cartesian product of two open sets is an element of the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
×s

19.3.13.5  Measures

Syntaxcmeas 24549 Extend class notation to include the class of measures.
measures

Definitiondf-meas 24550* Define a measure as a non-negative countably additive function over a sigma-algebra onto (Contributed by Thierry Arnoux, 10-Sep-2016.)
measures sigAlgebra Disj Σ*

Theoremmeasbase 24551 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra

Theoremmeasval 24552* The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
sigAlgebra measures Disj Σ*

Theoremismeas 24553* The property of being a measure (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
sigAlgebra measures Disj Σ*

Theoremisrnmeas 24554* The property of being a measure on an undefined base sigma algebra (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra Disj Σ*

Theoremdmmeas 24555 The domain of a measure is a sigma algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
measures sigAlgebra

Theoremmeasbasedom 24556 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

Theoremmeasfrge0 24557 A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasfn 24558 A measure is a function on its base sigma algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
measures

Theoremmeasvxrge0 24559 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasvnul 24560 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasge0 24561 A measure is non negative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
measures

Theoremmeasle0 24562 If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures

Theoremmeasvun 24563* The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures Disj Σ*

Theoremmeasxun2 24564 The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
measures

Theoremmeasun 24565 The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
measures

Theoremmeasvunilem 24566* Lemma for measvuni 24568 (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.)
measures Disj Σ*

Theoremmeasvunilem0 24567* Lemma for measvuni 24568. (Contributed by Thierry Arnoux, 6-Mar-2017.)
measures Disj Σ*

Theoremmeasvuni 24568* The measure of a countable disjoint union is the sum of the measures. This theorem uses a collection rather than a set of subsets of . (Contributed by Thierry Arnoux, 7-Mar-2017.)
measures Disj Σ*

Theoremmeasssd 24569 A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
measures

Theoremmeasunl 24570 A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures

Theoremmeasiuns 24571* The measure of the union of a collection of sets, expressed as the sum of a disjoint set. This is used as a lemma for both measiun 24572 and meascnbl 24573 (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
..^       measures              Σ* ..^

Theoremmeasiun 24572* A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
measures                            Σ*

Theoremmeascnbl 24573* A measure is continuous from below. Cf. volsup 19450. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
s        measures

Theoremmeasinblem 24574* Lemma for measinb 24575 (Contributed by Thierry Arnoux, 2-Jun-2017.)
measures Disj Σ*

Theoremmeasinb 24575* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

Theoremmeasres 24576 Building a measure restricted to a smaller sigma algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra measures

Theoremmeasinb2 24577* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

TheoremmeasdivcstOLD 24578* Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
measures /𝑒 measures

Theoremmeasdivcst 24579 Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (Revised by Thierry Arnoux, 30-Jan-2017.)
measures 𝑓/𝑐 /𝑒 measures

19.3.13.6  The counting measure

Theoremcntmeas 24580 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
sigAlgebra measures

Theorempwcntmeas 24581 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures

Theoremcntnevol 24582 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)

19.3.13.7  The Lebesgue measure - misc additions

Theoremvolss 24583 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)

Theoremunidmvol 24584 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theoremvoliune 24585 The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for for the measure of any set in the sum. Cf. ovoliun 19401 and voliun 19448 (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolfiniune 24586* The Lebesgue measure function is countably additive. This theorem is to volfiniun 19441 what voliune 24585 is to voliun 19448. (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolmeas 24587 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
measures

19.3.13.8  The 'almost everywhere' relation

Syntaxcae 24588 Extend class notation to include the 'almost everywhere' relation.
a.e.

Syntaxcfae 24589 Extend class notation to include the 'almost everywhere' builder.
~ a.e.

Definitiondf-ae 24590* Define 'almost everywhere' with regard to a measure . A property holds almost everywhere if the measure of the set where it does not hold has measure zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
a.e.

Theoremrelae 24591 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
a.e.

Theorembrae 24592 'almost everywhere' relation for a measure and a measurable set . (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theorembraew 24593* 'almost everywhere' relation for a measure and a property (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theoremtruae 24594* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures              a.e.

Theoremaean 24595* A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e. a.e. a.e.

Definitiondf-fae 24596* Define a builder for an 'almost everywhere' relation between functions, from relations between function values. In this definition, the range of and is enforced in order to ensure the resulting relation is a set. (Contributed by Thierry Arnoux, 22-Oct-2017.)
~ a.e. measures a.e.

Theoremfaeval 24597* Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures ~ a.e. a.e.

Theoremrelfae 24598 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures ~ a.e.

Theorembrfae 24599* 'almost everywhere' relation for two functions and with regard to the measure . (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures                     ~ a.e. a.e.

19.3.13.9  Measurable functions

Syntaxcmbfm 24600 Extend class notation with the measurable functions builder.
MblFnM

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