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

Theoremelsx 24501 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 24502 Extend class notation to include the class of measures.
measures

Definitiondf-meas 24503* 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 24504 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra

Theoremmeasval 24505* The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
sigAlgebra measures Disj Σ*

Theoremismeas 24506* 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 24507* The property of being a measure on an undefined base sigma algebra (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra Disj Σ*

Theoremdmmeas 24508 The domain of a measure is a sigma algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
measures sigAlgebra

Theoremmeasbasedom 24509 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

Theoremmeasfrge0 24510 A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasfn 24511 A measure is a function on its base sigma algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
measures

Theoremmeasvxrge0 24512 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasvnul 24513 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures

Theoremmeasge0 24514 A measure is non negative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
measures

Theoremmeasle0 24515 If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures

Theoremmeasvun 24516* The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
measures Disj Σ*

Theoremmeasxun2 24517 The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
measures

Theoremmeasun 24518 The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
measures

Theoremmeasvunilem 24519* Lemma for measvuni 24521 (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 24520* Lemma for measvuni 24521. (Contributed by Thierry Arnoux, 6-Mar-2017.)
measures Disj Σ*

Theoremmeasvuni 24521* 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 24522 A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
measures

Theoremmeasunl 24523 A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures

Theoremmeasiuns 24524* 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 24525 and meascnbl 24526 (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
..^       measures              Σ* ..^

Theoremmeasiun 24525* A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
measures                            Σ*

Theoremmeascnbl 24526* A measure is continuous from below. Cf. volsup 19403. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
s        measures

Theoremmeasinblem 24527* Lemma for measinb 24528 (Contributed by Thierry Arnoux, 2-Jun-2017.)
measures Disj Σ*

Theoremmeasinb 24528* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

Theoremmeasres 24529 Building a measure restricted to a smaller sigma algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures sigAlgebra measures

Theoremmeasinb2 24530* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures measures

TheoremmeasdivcstOLD 24531* 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 24532 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 24533 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
sigAlgebra measures

Theorempwcntmeas 24534 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures

Theoremcntnevol 24535 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)

19.3.13.7  The Lebesgue measure - misc additions

Theoremvolss 24536 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)

Theoremunidmvol 24537 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theoremvoliune 24538 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 19354 and voliun 19401 (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolfiniune 24539* The Lebesgue measure function is countably additive. This theorem is to volfiniun 19394 what voliune 24538 is to voliun 19401. (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolmeas 24540 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
measures

19.3.13.8  The 'almost everywhere' relation

Syntaxcae 24541 Extend class notation to include the 'almost everywhere' relation.
a.e.

Syntaxcfae 24542 Extend class notation to include the 'almost everywhere' builder.
~ a.e.

Definitiondf-ae 24543* 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 24544 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
a.e.

Theorembrae 24545 'almost everywhere' relation for a measure and a measurable set . (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theorembraew 24546* 'almost everywhere' relation for a measure and a property (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theoremtruae 24547* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures              a.e.

Theoremaean 24548* 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 24549* 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 24550* 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 24551 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures ~ a.e.

Theorembrfae 24552* '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 24553 Extend class notation with the measurable functions builder.
MblFnM

Definitiondf-mbfm 24554* Define the measurable function builder, which generates the set of measurable functions from a measurable space to another one. Here, the measurable spaces are given using their sigma algebra and , and the spaces themselves are recovered by and .

Note the similarities between the definition of measurable functions in measure theory, and of continuous functions in topology.

This is the definition for the generic measure theory. For the specific case of functions from to , see df-mbf 19465 (Contributed by Thierry Arnoux, 23-Jan-2017.)

MblFnM sigAlgebra sigAlgebra

Theoremismbfm 24555* The predicate " is a measurable function from the measurable space to the measurable space ". Cf. ismbf 19475 (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremelunirnmbfm 24556* The property of being a measurable function (Contributed by Thierry Arnoux, 23-Jan-2017.)
MblFnM sigAlgebra sigAlgebra

Theoremmbfmfun 24557 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
MblFnM

Theoremmbfmf 24558 A measurable function as a function with domain and codomain (Contributed by Thierry Arnoux, 25-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremisanmbfm 24559 The predicate to be a measurable function (Contributed by Thierry Arnoux, 30-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM       MblFnM

Theoremmbfmcnvima 24560 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremmbfmbfm 24561 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures              MblFnMsigaGen       MblFnM

Theoremmbfmcst 24562* A constant function is measurable. Cf. mbfconst 19480 (Contributed by Thierry Arnoux, 26-Jan-2017.)
sigAlgebra       sigAlgebra                     MblFnM

Theorem1stmbfm 24563 The first projection map is measurable with regard to the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
sigAlgebra       sigAlgebra       ×s MblFnM

Theorem2ndmbfm 24564 The second projection map is measurable with regard to the product sigma algebra (Contributed by Thierry Arnoux, 3-Jun-2017.)
sigAlgebra       sigAlgebra       ×s MblFnM

Theoremimambfm 24565* If the sigma-algebra in the range of a given function is generated by a collection of basic sets , then to check the measurability of that function, we need only consider inverse images of basic sets . (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra       sigaGen       MblFnM

Theoremcnmbfm 24566 A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
sigaGen       sigaGen       MblFnM

Theoremmbfmco 24567 The composition of two measurable functions is measurable. ( cf. cnmpt11 17648) (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM       MblFnM

Theoremmbfmco2 24568* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17650). (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM              MblFnM ×s

Theoremmbfmvolf 24569 Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
MblFnM𝔅

Theoremelmbfmvol2 24570 Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.)
MblFnM𝔅 MblFn

Theoremmbfmcnt 24571 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
MblFnM𝔅

19.3.13.10  Borel Algebra on ` ( RR X. RR ) `

Theorembr2base 24572* The base set for the generator of the Borel sigma algebra on is indeed . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅 𝔅

Theoremdya2ub 24573 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
logb

Theoremsxbrsigalem0 24574* The closed half-spaces of cover . (Contributed by Thierry Arnoux, 11-Oct-2017.)

Theoremsxbrsigalem3 24575* The sigma-algebra generated by the closed half-spaces of is a subset of the sigma-algebra generated by the closed sets of . (Contributed by Thierry Arnoux, 11-Oct-2017.)
sigaGen sigaGen

Theoremdya2iocival 24576* The function returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19445. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremdya2iocress 24577* Dyadic intervals are subsets of . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocbrsiga 24578* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅

Theoremdya2icobrsiga 24579* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
𝔅

Theoremdya2icoseg 24580* For any point and any closed below, opened above interval of centered on that point, there is a closed below opened above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.)
logb

Theoremdya2icoseg2 24581* For any point and any opened interval of containing that point, there is a closed below opened above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)

Theoremdya2iocrfn 24582* The function returning dyadic square covering for a given size has domain . (Contributed by Thierry Arnoux, 19-Sep-2017.)

Theoremdya2iocct 24583* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)

Theoremdya2iocnrect 24584* For any point of an opened rectangle in , there is a closed below opened above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)

Theoremdya2iocnei 24585* For any point of an open set of the usual topology on there is a closed below opened above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremdya2iocuni 24586* Every open set of is a union of closed below opened above dyadic rational rectangular subsets of . This union must be a countable union by dya2iocct 24583. (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocucvr 24587* The dyadic rectangular set collection covers . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremsxbrsigalem1 24588* The Borel algebra on is a subset of the sigma algebra generated by the dyadic closed below, opened above rectangular subsets of . This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4 (Contributed by Thierry Arnoux, 17-Sep-2017.)
sigaGen sigaGen

Theoremsxbrsigalem2 24589* The sigma-algebra generated by the dyadic closed below, opened above rectangular subsets of is a subset of the sigma algebra generated by the closed half-spaces of . The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)
sigaGen sigaGen

Theoremsxbrsigalem4 24590* The Borel algebra on is generated by the dyadic closed below, opened above rectangular subsets of . Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed below, opened above instead of opened below, closed above in the proof as they are ultimately generated by the floor function. (Contributed by Thierry Arnoux, 21-Sep-2017.)
sigaGen sigaGen

Theoremsxbrsigalem5 24591* First direction for sxbrsiga 24593. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsigalem6 24592 First direction for sxbrsiga 24593, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsiga 24593 The product sigma-algebra 𝔅 ×s 𝔅 is the Borel algebra on See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
𝔅 ×s 𝔅 sigaGen

19.3.14  Integration

19.3.14.1  Lebesgue integral - misc additions

Theoremitgeq12dv 24594* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)

19.3.14.2  Bochner integral

Syntaxcitgm 24595 Extend class notation with the (measure) Lebesgue integral.
itgm

Syntaxcsitm 24596 Extend class notation with the integral metric for simple functions.
sitm

Syntaxcsitg 24597 Extend class notation with the integral of simple functions.
sitg

Definitiondf-sitg 24598* Define the integral of simple functions from a measurable space to a generic space equipped with the right scalar product. will later be required to be a Banach space.

These simple functions are required to take finitely many different values: this is expressed by in the definition.

Moreover, for each , the pre-image is requested to be measurable, of finite measure.

In this definition, sigaGen is the Borel sigma-algebra on , and the functions range over the measurable functions over that Borel algebra.

Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)

sitg measures MblFnMsigaGen g RRHomScalar

Definitiondf-sitm 24599* Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take non-negative values in , the range structure for this integral is s . See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
sitm measures sitg sitg s sitg

Theoremsitgval 24600* Value of the simple function integral builder for a given space and measure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
sigaGen                     RRHomScalar              measures       sitg MblFnM g

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