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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembraew 24401* 'almost everywhere' relation for a measure and a property (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures a.e.

Theoremtruae 24402* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
measures              a.e.

Theoremaean 24403* 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 24404* 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 24405* 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 24406 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures ~ a.e.

Theorembrfae 24407* 'almost everywhere' relation for two functions and with regard to the measure . (Contributed by Thierry Arnoux, 22-Oct-2017.)
measures                     ~ a.e. a.e.

19.3.12.9  Measurable functions

Syntaxcmbfm 24408 Extend class notation with the measurable functions builder.
MblFnM

Definitiondf-mbfm 24409* 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 19393 (Contributed by Thierry Arnoux, 23-Jan-2017.)

MblFnM sigAlgebra sigAlgebra

Theoremismbfm 24410* The predicate " is a measurable function from the measurable space to the measurable space ". Cf. ismbf 19403 (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremelunirnmbfm 24411* The property of being a measurable function (Contributed by Thierry Arnoux, 23-Jan-2017.)
MblFnM sigAlgebra sigAlgebra

Theoremmbfmfun 24412 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
MblFnM

Theoremmbfmf 24413 A measurable function as a function with domain and codomain (Contributed by Thierry Arnoux, 25-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremisanmbfm 24414 The predicate to be a measurable function (Contributed by Thierry Arnoux, 30-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM       MblFnM

Theoremmbfmcnvima 24415 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
sigAlgebra       sigAlgebra       MblFnM

Theoremmbfmbfm 24416 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
measures              MblFnMsigaGen       MblFnM

Theoremmbfmcst 24417* A constant function is measurable. Cf. mbfconst 19408 (Contributed by Thierry Arnoux, 26-Jan-2017.)
sigAlgebra       sigAlgebra                     MblFnM

Theorem1stmbfm 24418 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 24419 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 24420* 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 24421 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 24422 The composition of two measurable functions is measurable. ( cf. cnmpt11 17630) (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM       MblFnM

Theoremmbfmco2 24423* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17632). (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra       sigAlgebra       sigAlgebra       MblFnM       MblFnM              MblFnM ×s

Theoremmbfmvolf 24424 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 24425 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 24426 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
MblFnM𝔅

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

Theorembr2base 24427* The base set for the generator of the Borel sigma algebra on is indeed . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅 𝔅

Theoremdya2ub 24428 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
logb

Theoremsxbrsigalem0 24429* The closed half-spaces of cover . (Contributed by Thierry Arnoux, 11-Oct-2017.)

Theoremsxbrsigalem3 24430* 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 24431* The function returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19373. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremdya2iocress 24432* Dyadic intervals are subsets of . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocbrsiga 24433* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝔅

Theoremdya2icobrsiga 24434* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
𝔅

Theoremdya2icoseg 24435* 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 24436* 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 24437* The function returning dyadic square covering for a given size has domain . (Contributed by Thierry Arnoux, 19-Sep-2017.)

Theoremdya2iocct 24438* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)

Theoremdya2iocnrect 24439* 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 24440* 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 24441* 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 24438. (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremdya2iocucvr 24442* The dyadic rectangular set collection covers . (Contributed by Thierry Arnoux, 18-Sep-2017.)

Theoremsxbrsigalem1 24443* 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 24444* 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 24445* 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 24446* First direction for sxbrsiga 24448. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsigalem6 24447 First direction for sxbrsiga 24448, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
sigaGen 𝔅 ×s 𝔅

Theoremsxbrsiga 24448 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

19.3.13  Integration

19.3.13.1  Lebesgue integral - misc additions

Theoremitgeq12dv 24449* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)

19.3.13.2  Bochner integral

Syntaxcitgm 24450 Extend class notation with the (measure) Lebesgue integral.
itgm

Syntaxcsitm 24451 Extend class notation with the integral metric for simple functions.
sitm

Syntaxcsitg 24452 Extend class notation with the integral of simple functions.
sitg

Syntaxcfndm 24453 Extend class notation with the fundamental sequences in the mean.
Fundm

Definitiondf-sitg 24454* Define the integral of simple functions for 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.

In this definition, sigaGenTopSet is the Borel sigma-algebra on , and the functions range over the measurable functions over that Borel algebra: this ensures that the preimage of the values of are measurable.

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

sitg measures MblFnMsigaGenTopSet g RRHomScalar

Definitiondf-sitm 24455* Define the integral metric for simple functions (Contributed by Thierry Arnoux, 22-Oct-2017.)
sitm measures sitg sitg sitg

Definitiondf-fndm 24456* TODO with the previous definiton, fundamental sequences shall be the Cauchy sequences for sitm: sitm

Define the 'fundamental in the mean' sequences, in the sense of the definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 21-Oct-2017.)

Fundm measures sitg sitg

Definitiondf-itgm 24457* Define the Bochner integral, following definition 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 19397. (Contributed by Thierry Arnoux, 21-Oct-2017.)
itgm measures Fundm TopSet TopSetsitg

19.3.14  Probability

19.3.14.1  Probability Theory

Syntaxcprb 24458 Extend class notation to include the class of probability measures.
Prob

Definitiondf-prob 24459 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 24460 The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
Prob measures

Theoremdomprobmeas 24461 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Prob measures

Theoremdomprobsiga 24462 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Prob sigAlgebra

Theoremprobtot 24463 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
Prob

Theoremprob01 24464 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobnul 24465 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremunveldomd 24466 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 24467 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 24468 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremprobcun 24469* 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 24470 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobdif 24471 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
Prob

Theoremprobinc 24472 A probabiliy law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
Prob

Theoremprobdsb 24473 The probability of the complement of a set. That is, the probability that the event does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
Prob

Theoremprobmeasd 24474 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob       measures

Theoremprobvalrnd 24475 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob

Theoremprobtotrnd 24476 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob

Theoremtotprobd 24477* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob                                   Disj        Σ*

Theoremtotprob 24478* Law of total probability (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob Disj Σ*

TheoremprobfinmeasbOLD 24479* Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
measures /𝑒 Prob

Theoremprobfinmeasb 24480 Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 31-Jan-2017.)
measures 𝑓/𝑐 /𝑒 Prob

Theoremprobmeasb 24481* Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures Prob

19.3.14.2  Conditional Probabilities

Syntaxccprob 24482 Extends class notation with the conditional probability builder.
cprob

Definitiondf-cndprob 24483* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
cprob Prob

Theoremcndprobval 24484 The value of the conditional probability , i.e. the probability for the event , given , under the probability law . (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobin 24485 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprob01 24486 The conditional probability has values in . (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobtot 24487 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobnul 24488 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobprob 24489* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob Prob

Theorembayesth 24490 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob cprob

19.3.14.3  Real Valued Random Variables

Syntaxcrrv 24491 Extend class notation with the class of real valued random variables.
rRndVar

Definitiondf-rrv 24492 In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
rRndVar Prob MblFnM𝔅

Theoremrrvmbfm 24493 A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar MblFnM𝔅

Theoremisrrvv 24494* Elementhood to the set of real-valued random variables with respect to the probability . (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar 𝔅

Theoremrrvvf 24495 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvfn 24496 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvdm 24497 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvrnss 24498 The range of a random variable as a subset of . (Contributed by Thierry Arnoux, 6-Feb-2017.)
Prob       rRndVar

Theoremrrvf2 24499 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvdmss 24500 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

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