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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsigaclcu2 24301* A sigma-algebra is closed under countable union - indexing on  NN (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A. k  e.  NN  A  e.  S )  -> 
 U_ k  e.  NN  A  e.  S )
 
Theoremsigaclfu2 24302* A sigma-algebra is closed under finite union - indexing on  ( 1..^ N ) (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A. k  e.  (
 1..^ N ) A  e.  S )  ->  U_ k  e.  ( 1..^ N ) A  e.  S )
 
Theoremsigaclcu3 24303* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   &    |-  ( ( ph  /\  k  e.  N )  ->  A  e.  S )   =>    |-  ( ph  ->  U_ k  e.  N  A  e.  S )
 
Theoremissgon 24304 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.)
 |-  ( S  e.  (sigAlgebra `  O ) 
 <->  ( S  e.  U. ran sigAlgebra  /\  O  =  U. S ) )
 
Theoremsgon 24305 A sigma alebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
 
Theoremelsigass 24306 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S ) 
 ->  A  C_  U. S )
 
Theoremelrnsiga 24307 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( S  e.  (sigAlgebra `  O )  ->  S  e.  U. ran sigAlgebra )
 
Theoremisrnsigau 24308* The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  ( S  C_  ~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
 ->  U. x  e.  S ) ) ) )
 
Theoremunielsiga 24309 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
 |-  ( S  e.  U. ran sigAlgebra  ->  U. S  e.  S )
 
Theoremdmvlsiga 24310 Lebesgue-measurable subsets of  RR form a sigma-algebra (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
 |-  dom  vol 
 e.  (sigAlgebra `  RR )
 
Theorempwsiga 24311 Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
 |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O ) )
 
Theoremprsiga 24312 The smallest possible sigma-algebra containing  O (Contributed by Thierry Arnoux, 13-Sep-2016.)
 |-  ( O  e.  V  ->  { (/) ,  O }  e.  (sigAlgebra `
  O ) )
 
Theoremsigaclci 24313 A sigma-algebra is closed under countable intersection. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
 |-  (
 ( ( S  e.  U.
 ran sigAlgebra  /\  A  e.  ~P S )  /\  ( A  ~<_ 
 om  /\  A  =/=  (/) ) )  ->  |^| A  e.  S )
 
Theoremdifelsiga 24314 A sigma algebra is closed under set difference. (Contributed by Thierry Arnoux, 13-Sep-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  \  B )  e.  S )
 
Theoremunelsiga 24315 A sigma algebra is closed under set union. (Contributed by Thierry Arnoux, 13-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  u.  B )  e.  S )
 
Theoreminelsiga 24316 A sigma algebra is closed under set intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  i^i  B )  e.  S )
 
Theoremsigainb 24317 Building a sigma algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S ) 
 ->  ( S  i^i  ~P A )  e.  (sigAlgebra `  A ) )
 
Theoreminsiga 24318 The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  (
 ( A  =/=  (/)  /\  A  e.  ~P (sigAlgebra `  O ) ) 
 ->  |^| A  e.  (sigAlgebra `  O ) )
 
19.3.12.2  Generated Sigma-Algebra
 
Syntaxcsigagen 24319 Extend class notation to include the sigma-algebra generator.
 class sigaGen
 
Definitiondf-sigagen 24320* 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  =  ( x  e.  _V  |->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s } )
 
Theoremsigagenval 24321* Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
 
Theoremsigagensiga 24322 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
 U. A ) )
 
Theoremsgsiga 24323 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  (sigaGen `  A )  e.  U. ran sigAlgebra )
 
Theoremunisg 24324 The sigma algebra generated by a collection  A is a sigma algebra on  U. A. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )
 
Theoremdmsigagen 24325 A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  dom sigaGen  =  _V
 
Theoremsssigagen 24326 A set is a subset of the sigma algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( A  e.  V  ->  A 
 C_  (sigaGen `  A )
 )
 
Theoremsssigagen2 24327 A subset of the generating set is also a subset of the generated sigma algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  (
 ( A  e.  V  /\  B  C_  A )  ->  B  C_  (sigaGen `  A ) )
 
Theoremelsigagen 24328 Any element of set is also an element of the sigma algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  A ) 
 ->  B  e.  (sigaGen `  A ) )
 
Theoremelsigagen2 24329 Any countable union of elements of a set is also in the sigma algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  (
 ( A  e.  V  /\  B  C_  A  /\  B 
 ~<_  om )  ->  U. B  e.  (sigaGen `  A )
 )
 
Theoremsigagenss 24330 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  B. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  (
 ( S  e.  (sigAlgebra ` 
 U. A )  /\  A  C_  S )  ->  (sigaGen `  A )  C_  S )
 
Theoremsigagenss2 24331 Sufficient condition for inclusion of sigma algebra. This is used to prove equality of sigma algebra. (Contributed by Thierry Arnoux, 10-Oct-2017.)
 |-  (
 ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  A )  C_  (sigaGen `  B ) )
 
Theoremsigagenid 24332 The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( S  e.  U. ran sigAlgebra  ->  (sigaGen `  S )  =  S )
 
19.3.12.3  The Borel algebra on the real numbers
 
Syntaxcbrsiga 24333 The Borel Algebra on real numbers, usually a gothic B
 class 𝔅
 
Definitiondf-brsiga 24334 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  J is the sigma-algebra generated by 
J,  (sigaGen `  J
), so there is no need to introduce a special constant function for Borel sigma Algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  =  (sigaGen `  ( topGen `
  ran  (,) ) )
 
Theorembrsiga 24335 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigaGen " Top )
 
Theorembrsigarn 24336 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigAlgebra `  RR )
 
Theorembrsigasspwrn 24337 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |- 𝔅 
 C_  ~P RR
 
Theoremunibrsiga 24338 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  U.𝔅  =  RR
 
Theoremcldssbrsiga 24339 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
 
19.3.12.4  Product Sigma-Algebra
 
Syntaxcsx 24340 Extend class notation with the product sigma-algebra operation.
 class ×s
 
Definitiondf-sx 24341* Define the product sigma-algebra operation, analogue to df-tx 17517. (Contributed by Thierry Arnoux, 1-Jun-2017.)
 |- ×s  =  (
 s  e.  _V ,  t  e.  _V  |->  (sigaGen `  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y
 ) ) ) )
 
Theoremsxval 24342* Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
 |-  A  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )   =>    |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  A ) )
 
Theoremsxsiga 24343 A product sigma-algebra is a sigma-algebra (Contributed by Thierry Arnoux, 1-Jun-2017.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra ) 
 ->  ( S ×s  T )  e.  U. ran sigAlgebra )
 
Theoremsxsigon 24344 A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra ) 
 ->  ( S ×s  T )  e.  (sigAlgebra `  ( U. S  X.  U. T ) ) )
 
Theoremsxuni 24345 The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra ) 
 ->  ( U. S  X.  U. T )  =  U. ( S ×s  T ) )
 
Theoremelsx 24346 The cartesian product of two open sets is an element of the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  (
 ( ( S  e.  V  /\  T  e.  W )  /\  ( A  e.  S  /\  B  e.  T ) )  ->  ( A  X.  B )  e.  ( S ×s  T ) )
 
19.3.12.5  Measures
 
Syntaxcmeas 24347 Extend class notation to include the class of measures.
 class measures
 
Definitiondf-meas 24348* Define a measure as a non-negative countably additive function over a sigma-algebra onto  ( 0 [,]  +oo ) (Contributed by Thierry Arnoux, 10-Sep-2016.)
 |- measures  =  ( s  e.  U. ran sigAlgebra  |->  { m  |  ( m : s --> ( 0 [,]  +oo )  /\  ( m `  (/) )  =  0 
 /\  A. x  e.  ~P  s ( ( x  ~<_ 
 om  /\ Disj  y  e.  x y )  ->  ( m `
  U. x )  = Σ* y  e.  x ( m `
  y ) ) ) } )
 
Theoremmeasbase 24349 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  S  e.  U. ran sigAlgebra )
 
Theoremmeasval 24350* The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  (measures `  S )  =  { m  |  ( m : S --> ( 0 [,]  +oo )  /\  ( m `
  (/) )  =  0 
 /\  A. x  e.  ~P  S ( ( x  ~<_ 
 om  /\ Disj  y  e.  x y )  ->  ( m `
  U. x )  = Σ* y  e.  x ( m `
  y ) ) ) } )
 
Theoremismeas 24351* The property of being a measure (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  S ) 
 <->  ( M : S --> ( 0 [,]  +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om 
 /\ Disj  y  e.  x y )  ->  ( M ` 
 U. x )  = Σ* y  e.  x ( M `
  y ) ) ) ) )
 
Theoremisrnmeas 24352* The property of being a measure on an undefined base sigma algebra (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( M  e.  U. ran measures  ->  ( dom  M  e.  U. ran sigAlgebra  /\  ( M : dom  M --> ( 0 [,]  +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om 
 /\ Disj  y  e.  x y )  ->  ( M ` 
 U. x )  = Σ* y  e.  x ( M `
  y ) ) ) ) )
 
Theoremmeasbasedom 24353 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( M  e.  U. ran measures  <->  M  e.  (measures ` 
 dom  M ) )
 
Theoremmeasfrge0 24354 A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  M : S --> ( 0 [,]  +oo ) )
 
Theoremmeasfn 24355 A measure is a function on its base sigma algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( M  e.  (measures `  S )  ->  M  Fn  S )
 
Theoremmeasvxrge0 24356 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S )  ->  ( M `  A )  e.  ( 0 [,]  +oo ) )
 
Theoremmeasvnul 24357 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  ( M `  (/) )  =  0 )
 
Theoremmeasle0 24358 If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `
  A )  <_ 
 0 )  ->  ( M `  A )  =  0 )
 
Theoremmeasvun 24359* The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  ~P S  /\  ( A 
 ~<_  om  /\ Disj  x  e.  A x ) )  ->  ( M `  U. A )  = Σ* x  e.  A ( M `  x ) )
 
Theoremmeasxun2 24360 The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( A  e.  S  /\  B  e.  S )  /\  B  C_  A )  ->  ( M `  A )  =  ( ( M `  B ) + e ( M `  ( A  \  B ) ) ) )
 
Theoremmeasun 24361 The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( A  e.  S  /\  B  e.  S )  /\  ( A  i^i  B )  =  (/) )  ->  ( M `  ( A  u.  B ) )  =  ( ( M `
  A ) + e ( M `  B ) ) )
 
Theoremmeasvunilem 24362* Lemma for measvuni 24364 (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.)
 |-  F/_ x A   =>    |-  ( ( M  e.  (measures `  S )  /\  A. x  e.  A  B  e.  ( S  \  { (/)
 } )  /\  ( A 
 ~<_  om  /\ Disj  x  e.  A B ) )  ->  ( M `  U_ x  e.  A  B )  = Σ* x  e.  A ( M `
  B ) )
 
Theoremmeasvunilem0 24363* Lemma for measvuni 24364. (Contributed by Thierry Arnoux, 6-Mar-2017.)
 |-  F/_ x A   =>    |-  ( ( M  e.  (measures `  S )  /\  A. x  e.  A  B  e.  { (/) }  /\  ( A 
 ~<_  om  /\ Disj  x  e.  A B ) )  ->  ( M `  U_ x  e.  A  B )  = Σ* x  e.  A ( M `
  B ) )
 
Theoremmeasvuni 24364* 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  S. (Contributed by Thierry Arnoux, 7-Mar-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  A. x  e.  A  B  e.  S  /\  ( A  ~<_ 
 om  /\ Disj  x  e.  A B ) )  ->  ( M `  U_ x  e.  A  B )  = Σ* x  e.  A ( M `
  B ) )
 
Theoremmeasssd 24365 A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  ( ph  ->  M  e.  (measures `  S ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  ( M `  A ) 
 <_  ( M `  B ) )
 
Theoremmeasunl 24366 A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  ( ph  ->  M  e.  (measures `  S ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   =>    |-  ( ph  ->  ( M `  ( A  u.  B ) )  <_  ( ( M `  A ) + e
 ( M `  B ) ) )
 
Theoremmeasiuns 24367* 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 24368 and meascnbl 24369 (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ I ) ) )   &    |-  ( ph  ->  M  e.  (measures `  S )
 )   &    |-  ( ( ph  /\  n  e.  N )  ->  A  e.  S )   =>    |-  ( ph  ->  ( M `  U_ n  e.  N  A )  = Σ* n  e.  N ( M `
  ( A  \  U_ k  e.  ( 1..^ n ) B ) ) )
 
Theoremmeasiun 24368* A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
 |-  ( ph  ->  M  e.  (measures `  S ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  n  e.  NN )  ->  B  e.  S )   &    |-  ( ph  ->  A 
 C_  U_ n  e.  NN  B )   =>    |-  ( ph  ->  ( M `  A )  <_ Σ* n  e.  NN ( M `  B ) )
 
Theoremmeascnbl 24369* A measure is continuous from below. Cf. volsup 19319. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  ( ph  ->  M  e.  (measures `  S ) )   &    |-  ( ph  ->  F : NN
 --> S )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  C_  ( F `  ( n  +  1
 ) ) )   =>    |-  ( ph  ->  ( M  o.  F ) ( ~~> t `  J ) ( M `  U.
 ran  F ) )
 
Theoremmeasinblem 24370* Lemma for measinb 24371 (Contributed by Thierry Arnoux, 2-Jun-2017.)
 |-  (
 ( ( ( M  e.  (measures `  S )  /\  A  e.  S ) 
 /\  B  e.  ~P S )  /\  ( B  ~<_ 
 om  /\ Disj  x  e.  B x ) )  ->  ( M `  ( U. B  i^i  A ) )  = Σ* x  e.  B ( M `  ( x  i^i  A ) ) )
 
Theoremmeasinb 24371* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S )  ->  ( x  e.  S  |->  ( M `
  ( x  i^i  A ) ) )  e.  (measures `  S )
 )
 
Theoremmeasres 24372 Building a measure restricted to a smaller sigma algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  ->  ( M  |`  T )  e.  (measures `  T ) )
 
Theoremmeasinb2 24373* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S )  ->  ( x  e.  ( S  i^i  ~P A )  |->  ( M `  ( x  i^i  A ) ) )  e.  (measures `  ( S  i^i  ~P A ) ) )
 
TheoremmeasdivcstOLD 24374* 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.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
 
Theoremmeasdivcst 24375 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.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  e.  (measures `  S )
 )
 
19.3.12.6  The counting measure
 
Theoremcntmeas 24376 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  ( #  |`  S )  e.  (measures `  S ) )
 
Theorempwcntmeas 24377 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( O  e.  V  ->  ( #  |`  ~P O )  e.  (measures `  ~P O ) )
 
Theoremcntnevol 24378 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( #  |`  ~P O )  =/= 
 vol
 
19.3.12.7  The Lebesgue measure - misc additions
 
Theoremvolss 24379 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
 |-  (
 ( A  e.  dom  vol  /\  B  e.  dom  vol  /\  A  C_  B )  ->  ( vol `  A )  <_  ( vol `  B ) )
 
Theoremunidmvol 24380 The union of the Lebesgue measurable sets is  RR. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  U. dom  vol 
 =  RR
 
Theoremvoliune 24381 The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for  +oo for the measure of any set in the sum. Cf. ovoliun 19270 and voliun 19317 (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  (
 ( A. n  e.  NN  A  e.  dom  vol  /\ Disj  n  e.  NN A )  ->  ( vol `  U_ n  e. 
 NN  A )  = Σ* n  e.  NN ( vol `  A ) )
 
Theoremvolfiniune 24382* The Lebesgue measure function is countably additive. This theorem is to volfiniun 19310 what voliune 24381 is to voliun 19317. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  (
 ( A  e.  Fin  /\ 
 A. n  e.  A  B  e.  dom  vol  /\ Disj  n  e.  A B )  ->  ( vol `  U_ n  e.  A  B )  = Σ* n  e.  A ( vol `  B ) )
 
Theoremvolmeas 24383 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  vol  e.  (measures `  dom  vol )
 
19.3.12.8  The 'almost everywhere' relation
 
Syntaxcae 24384 Extend class notation to include the 'almost everywhere' relation.
 class a.e.
 
Syntaxcfae 24385 Extend class notation to include the 'almost everywhere' builder.
 class ~ a.e.
 
Definitiondf-ae 24386* Define 'almost everywhere' with regard to a measure  M. 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.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
 ) )  =  0 }
 
Theoremrelae 24387 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  Rel a.e.
 
Theorembrae 24388 'almost everywhere' relation for a measure and a measurable set  A. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  (
 ( M  e.  U. ran measures 
 /\  A  e.  dom  M )  ->  ( Aa.e. M  <-> 
 ( M `  ( U. dom  M  \  A ) )  =  0
 ) )
 
Theorembraew 24389* 'almost everywhere' relation for a measure  M and a property  ph (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  U. dom  M  =  O   =>    |-  ( M  e.  U. ran measures 
 ->  ( { x  e.  O  |  ph }a.e. M  <->  ( M `  { x  e.  O  |  -.  ph } )  =  0 ) )
 
Theoremtruae 24390* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  U. dom  M  =  O   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  { x  e.  O  |  ps }a.e. M )
 
Theoremaean 24391* A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  U. dom  M  =  O   =>    |-  ( ( M  e.  U.
 ran measures  /\  { x  e.  O  |  -.  ph }  e.  dom  M  /\  { x  e.  O  |  -.  ps }  e.  dom  M )  ->  ( { x  e.  O  |  ( ph  /\  ps ) }a.e. M  <->  ( { x  e.  O  |  ph }a.e. M  /\  { x  e.  O  |  ps }a.e. M ) ) )
 
Definitiondf-fae 24392* Define a builder for an 'almost everywhere' relation between functions, from relations between function values. In this definition, the range of  f and  g is enforced in order to ensure the resulting relation is a set. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |- ~ a.e.  =  ( r  e.  _V ,  m  e.  U. ran measures  |->  { <. f ,  g >.  |  ( ( f  e.  ( dom  r  ^m  U. dom  m )  /\  g  e.  ( dom  r  ^m  U.
 dom  m ) ) 
 /\  { x  e.  U. dom  m  |  ( f `
  x ) r ( g `  x ) }a.e. m ) }
 )
 
Theoremfaeval 24393* Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  (
 ( R  e.  _V  /\  M  e.  U. ran measures ) 
 ->  ( R~ a.e. M )  =  { <. f ,  g >.  |  (
 ( f  e.  ( dom  R  ^m  U. dom  M )  /\  g  e.  ( dom  R  ^m  U.
 dom  M ) )  /\  { x  e.  U. dom  M  |  ( f `  x ) R ( g `  x ) }a.e. M ) }
 )
 
Theoremrelfae 24394 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  (
 ( R  e.  _V  /\  M  e.  U. ran measures ) 
 ->  Rel  ( R~ a.e. M ) )
 
Theorembrfae 24395* 'almost everywhere' relation for two functions  F and 
G with regard to the measure  M. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  dom  R  =  D   &    |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  ( D  ^m  U.
 dom  M ) )   &    |-  ( ph  ->  G  e.  ( D  ^m  U. dom  M ) )   =>    |-  ( ph  ->  ( F ( R~ a.e. M ) G  <->  { x  e.  U. dom  M  |  ( F `
  x ) R ( G `  x ) }a.e. M ) )
 
19.3.12.9  Measurable functions
 
Syntaxcmbfm 24396 Extend class notation with the measurable functions builder.
 class MblFnM
 
Definitiondf-mbfm 24397* 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  s and  t, and the spaces themselves are recovered by  U. s and  U. t.

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  RR to  CC, see df-mbf 19381 (Contributed by Thierry Arnoux, 23-Jan-2017.)

 |- MblFnM  =  ( s  e.  U. ran sigAlgebra ,  t  e.  U. ran sigAlgebra  |->  { f  e.  ( U. t  ^m  U. s )  |  A. x  e.  t  ( `' f " x )  e.  s } )
 
Theoremismbfm 24398* The predicate " F is a measurable function from the measurable space  S to the measurable space  T". Cf. ismbf 19391 (Contributed by Thierry Arnoux, 23-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( F  e.  ( SMblFnM
 T )  <->  ( F  e.  ( U. T  ^m  U. S )  /\  A. x  e.  T  ( `' F " x )  e.  S ) ) )
 
Theoremelunirnmbfm 24399* The property of being a measurable function (Contributed by Thierry Arnoux, 23-Jan-2017.)
 |-  ( F  e.  U. ran MblFnM  <->  E. s  e.  U. ran sigAlgebra E. t  e.  U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s ) 
 /\  A. x  e.  t  ( `' F " x )  e.  s ) )
 
Theoremmbfmfun 24400 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( ph  ->  F  e.  U. ran MblFnM )   =>    |-  ( ph  ->  Fun  F )
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