HomeHome Metamath Proof Explorer
Theorem List (p. 246 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22421)
  Hilbert Space Explorer  Hilbert Space Explorer
(22422-23944)
  Users' Mathboxes  Users' Mathboxes
(23945-32762)
 

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.)
 |-  F/_ k A   =>    |-  ( ( S  e.  U.
 ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U_ k  e.  A  B  e.  S )
 
Theoremsigaclfu 24502 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  e.  Fin )  ->  U. A  e.  S )
 
Theoremsigaclcu2 24503* 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 24504* 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 24505* 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 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.)
 |-  ( S  e.  (sigAlgebra `  O ) 
 <->  ( S  e.  U. ran sigAlgebra  /\  O  =  U. S ) )
 
Theoremsgon 24507 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 24508 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 24509 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 24510* 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 24511 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 24512 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 24513 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 24514 The smallest possible sigma-algebra containing  O (Contributed by Thierry Arnoux, 13-Sep-2016.)
 |-  ( O  e.  V  ->  { (/) ,  O }  e.  (sigAlgebra `
  O ) )
 
Theoremsigaclci 24515 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 24516 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 24517 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 24518 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 24519 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 24520 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 ) )
 
19.3.13.2  Generated Sigma-Algebra
 
Syntaxcsigagen 24521 Extend class notation to include the sigma-algebra generator.
 class 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  =  ( x  e.  _V  |->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s } )
 
Theoremsigagenval 24523* 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 24524 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 24525 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 24526 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 24527 A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  dom sigaGen  =  _V
 
Theoremsssigagen 24528 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 24529 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 24530 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 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.)
 |-  (
 ( A  e.  V  /\  B  C_  A  /\  B 
 ~<_  om )  ->  U. B  e.  (sigaGen `  A )
 )
 
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  B. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  (
 ( S  e.  (sigAlgebra ` 
 U. A )  /\  A  C_  S )  ->  (sigaGen `  A )  C_  S )
 
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.)
 |-  (
 ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  A )  C_  (sigaGen `  B ) )
 
Theoremsigagenid 24534 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 )
 
19.3.13.3  The Borel algebra on the real numbers
 
Syntaxcbrsiga 24535 The Borel Algebra on real numbers, usually a gothic B
 class 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  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 24537 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigaGen " Top )
 
Theorembrsigarn 24538 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigAlgebra `  RR )
 
Theorembrsigasspwrn 24539 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |- 𝔅 
 C_  ~P RR
 
Theoremunibrsiga 24540 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  U.𝔅  =  RR
 
Theoremcldssbrsiga 24541 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 ) )
 
19.3.13.4  Product Sigma-Algebra
 
Syntaxcsx 24542 Extend class notation with the product sigma-algebra operation.
 class ×s
 
Definitiondf-sx 24543* Define the product sigma-algebra operation, analogue to df-tx 17594. (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 24544* 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 24545 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 24546 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 24547 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 24548 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 ) )
 
19.3.13.5  Measures
 
Syntaxcmeas 24549 Extend class notation to include the class of measures.
 class measures
 
Definitiondf-meas 24550* 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 24551 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 24552* 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 24553* 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 24554* 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 ) ) ) ) )
 
Theoremdmmeas 24555 The domain of a measure is a sigma algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
 
Theoremmeasbasedom 24556 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 24557 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 24558 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 24559 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 24560 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  ( M `  (/) )  =  0 )
 
Theoremmeasge0 24561 A measure is non negative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S )  ->  0  <_  ( M `  A ) )
 
Theoremmeasle0 24562 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 24563* 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 24564 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 24565 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 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.)
 |-  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 24567* Lemma for measvuni 24568. (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 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  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 24569 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 24570 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 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.)
 |-  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 24572* 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 24573* A measure is continuous from below. Cf. volsup 19450. (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 24574* Lemma for measinb 24575 (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 24575* 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 24576 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 24577* 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 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.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
 
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.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  e.  (measures `  S )
 )
 
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.)
 |-  ( S  e.  U. ran sigAlgebra  ->  ( #  |`  S )  e.  (measures `  S ) )
 
Theorempwcntmeas 24581 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 24582 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( #  |`  ~P O )  =/= 
 vol
 
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.)
 |-  (
 ( A  e.  dom  vol  /\  B  e.  dom  vol  /\  A  C_  B )  ->  ( vol `  A )  <_  ( vol `  B ) )
 
Theoremunidmvol 24584 The union of the Lebesgue measurable sets is  RR. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  U. dom  vol 
 =  RR
 
Theoremvoliune 24585 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 19401 and voliun 19448 (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 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.)
 |-  (
 ( 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 24587 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  vol  e.  (measures `  dom  vol )
 
19.3.13.8  The 'almost everywhere' relation
 
Syntaxcae 24588 Extend class notation to include the 'almost everywhere' relation.
 class a.e.
 
Syntaxcfae 24589 Extend class notation to include the 'almost everywhere' builder.
 class ~ a.e.
 
Definitiondf-ae 24590* 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 24591 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  Rel a.e.
 
Theorembrae 24592 '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 24593* '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 24594* 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 24595* 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 24596* 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 24597* 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 24598 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 24599* '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 ) )
 
19.3.13.9  Measurable functions
 
Syntaxcmbfm 24600 Extend class notation with the measurable functions builder.
 class MblFnM
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32762
  Copyright terms: Public domain < Previous  Next >