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

Theoremmeasdivcst 24501 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 24502 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
sigAlgebra measures

Theorempwcntmeas 24503 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures

Theoremcntnevol 24504 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)

19.3.13.7  The Lebesgue measure - misc additions

Theoremvolss 24505 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)

Theoremunidmvol 24506 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theoremvoliune 24507 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 19340 and voliun 19387 (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolfiniune 24508* The Lebesgue measure function is countably additive. This theorem is to volfiniun 19380 what voliune 24507 is to voliun 19387. (Contributed by Thierry Arnoux, 16-Oct-2017.)
Disj Σ*

Theoremvolmeas 24509 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
measures

19.3.13.8  The 'almost everywhere' relation

Syntaxcae 24510 Extend class notation to include the 'almost everywhere' relation.
a.e.

Syntaxcfae 24511 Extend class notation to include the 'almost everywhere' builder.
~ a.e.

Definitiondf-ae 24512* 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 24513 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
a.e.

Theorembrae 24514 'almost everywhere' relation for a measure and a measurable set . (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theorembraew 24515* 'almost everywhere' relation for a measure and a property (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theoremtruae 24516* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures              a.e.

Theoremaean 24517* 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 24518* 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 24519* 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 24520 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures ~ a.e.

Theorembrfae 24521* '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 24522 Extend class notation with the measurable functions builder.
MblFnM

Definitiondf-mbfm 24523* 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 19451 (Contributed by Thierry Arnoux, 23-Jan-2017.)

MblFnM sigAlgebra sigAlgebra

Theoremismbfm 24524* The predicate " is a measurable function from the measurable space to the measurable space ". Cf. ismbf 19461 (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremelunirnmbfm 24525* The property of being a measurable function (Contributed by Thierry Arnoux, 23-Jan-2017.)
MblFnM sigAlgebra sigAlgebra

Theoremmbfmfun 24526 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
MblFnM

Theoremmbfmf 24527 A measurable function as a function with domain and codomain (Contributed by Thierry Arnoux, 25-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremisanmbfm 24528 The predicate to be a measurable function (Contributed by Thierry Arnoux, 30-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM       MblFnM

Theoremmbfmcnvima 24529 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremmbfmbfm 24530 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures              MblFnMsigaGen       MblFnM

Theoremmbfmcst 24531* A constant function is measurable. Cf. mbfconst 19466 (Contributed by Thierry Arnoux, 26-Jan-2017.)
sigAlgebra       sigAlgebra                     MblFnM

Theorem1stmbfm 24532 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 24533 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 24534* 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 24535 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 24536 The composition of two measurable functions is measurable. ( cf. cnmpt11 17634) (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM       MblFnM

Theoremmbfmco2 24537* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17636). (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM              MblFnM ×s

Theoremmbfmvolf 24538 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 24539 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 24540 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 24541* The base set for the generator of the Borel sigma algebra on is indeed . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅 𝔅

Theoremdya2ub 24542 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
logb

Theoremsxbrsigalem0 24543* The closed half-spaces of cover . (Contributed by Thierry Arnoux, 11-Oct-2017.)

Theoremsxbrsigalem3 24544* 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 24545* The function returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19431. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremdya2iocress 24546* Dyadic intervals are subsets of . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocbrsiga 24547* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅

Theoremdya2icobrsiga 24548* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
𝔅

Theoremdya2icoseg 24549* 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 24550* 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 24551* The function returning dyadic square covering for a given size has domain . (Contributed by Thierry Arnoux, 19-Sep-2017.)

Theoremdya2iocct 24552* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)

Theoremdya2iocnrect 24553* 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 24554* 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 24555* 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 24552. (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocucvr 24556* The dyadic rectangular set collection covers . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremsxbrsigalem1 24557* 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 24558* 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 24559* 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 24560* First direction for sxbrsiga 24562. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsigalem6 24561 First direction for sxbrsiga 24562, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsiga 24562 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 24563* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)

19.3.14.2  Bochner integral

Syntaxcitgm 24564 Extend class notation with the (measure) Lebesgue integral.
itgm

Syntaxcsitm 24565 Extend class notation with the integral metric for simple functions.
sitm

Syntaxcsitg 24566 Extend class notation with the integral of simple functions.
sitg

Definitiondf-sitg 24567* 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 24568* 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 24569* 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

Theoremissibf 24570* The predicate " is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg MblFnM

Theoremsibf0 24571 The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
sigaGen                     RRHomScalar              measures                     sitg

Theoremsibfmbl 24572 A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg       MblFnM

Theoremsibff 24573 A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg

Theoremsibfrn 24574 A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg

Theoremsibfima 24575 Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg

Theoremsibfof 24576 Applying function operations on simple functions results in simple functions with regard to the the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
sigaGen                     RRHomScalar              measures       sitg                            sitg                            sitg

Theoremsitgfval 24577* Value of the Bochner integral for a simple function . (Contributed by Thierry Arnoux, 30-Jan-2018.)
sigaGen                     RRHomScalar              measures       sitg       sitg g

Theoremsitgclg 24578* Closure of the Bochner integral on a simple functions. This version is very generic, thus the many hypothesis. See sitgclbn 24579 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg       Scalar                     CMnd              NrmRing       Mod NrmMod       chr               CUnifSp              UnifSt metUnif              sitg

Theoremsitgclbn 24579 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg       Scalar              Ban       CUnifSp              UnifSt metUnif       Mod NrmMod       chr        sitg

Theoremsitgclcn 24580 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg       Ban       Scalar fld       sitg

Theoremsitgclre 24581 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg       Ban       Scalar flds        sitg

Theoremsitgf 24582* The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
sigaGen                     RRHomScalar              measures       sitg sitg        sitg sitg

Theoremsitmval 24583* Value of the simple function integral metric for a given space and measure . (Contributed by Thierry Arnoux, 30-Jan-2018.)
measures       sitm sitg sitg s sitg

Theoremsitmfval 24584 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
measures       sitg       sitg       sitm s sitg

Theoremsitmcl 24585 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
measures       sitg       sitg       sitm

Definitiondf-itgm 24586* Define the Bochner integral as the extension by continuity of the Bochnel integral for simple functions.

Bogachev first defines 'fundamental in the mean' sequences, in definition 2.3.1 of [Bogachev] p. 116, and notes that those are actually Cauchy sequences for the pseudometric sitm.

He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 19455.

(Contributed by Thierry Arnoux, 13-Feb-2018.)

itgm measures metUnifsitmCnExtUnifStsitg

19.3.15  Probability

19.3.15.1  Probability Theory

Syntaxcprb 24587 Extend class notation to include the class of probability measures.
Prob

Definitiondf-prob 24588 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
Prob measures

Theoremelprob 24589 The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
Prob measures

Theoremdomprobmeas 24590 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Prob measures

Theoremdomprobsiga 24591 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Prob sigAlgebra

Theoremprobtot 24592 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
Prob

Theoremprob01 24593 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobnul 24594 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremunveldomd 24595 The universe is an element of the domain of the probability, the universe (entire probability space) being in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremunveldom 24596 The universe is an element of the domain of the probability, the universe (entire probability space) being in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremnuleldmp 24597 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremprobcun 24598* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the construct cannot be used as it can handle infinite indexing set only if they are subsets of , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob Disj Σ*

Theoremprobun 24599 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobdif 24600 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
Prob

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