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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.3.13.3  The Borel algebra on the real numbers
 
Syntaxcbrsiga 23801 The Borel Algebra on real numbers, usually a gothic B
 class 𝔅
 
Definitiondf-brsiga 23802 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 23803 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigaGen " Top )
 
Theorembrsigarn 23804 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigAlgebra `  RR )
 
Theorembrsigasspwrn 23805 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |- 𝔅 
 C_  ~P RR
 
Theoremunibrsiga 23806 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  U.𝔅  =  RR
 
Theoremcldssbrsiga 23807 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 ) )
 
18.3.13.4  Product Sigma-Algebra
 
Syntaxcsx 23808 Extend class notation with the product sigma-algebra operation.
 class ×s
 
Definitiondf-sx 23809* Define the product sigma-algebra operation, analogue to df-tx 17357. (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 23810* 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 23811 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 23812 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 23813 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 23814 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 ) )
 
18.3.13.5  Measures
 
Syntaxcmeas 23815 Extend class notation to include the class of measures.
 class measures
 
Definitiondf-meas 23816* 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 23817 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 23818* 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 23819* 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 23820* 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 23821 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 23822 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 23823 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 23824 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 23825 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  ( M `  (/) )  =  0 )
 
Theoremmeasle0 23826 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 23827* 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 23828 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 23829 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 23830* Lemma for measvuni 23832 (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 23831* Lemma for measvuni 23832. (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 23832* 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 23833 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 23834 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 23835* 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 23836 and meascnbl 23837 (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 23836* 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 23837* A measure is continuous from below. Cf. volsup 19011. (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 23838* Lemma for measinb 23839 (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 23839* 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 23840 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 23841* 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 23842* Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
 
Theoremmeasdivcst 23843 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 )
 )
 
18.3.13.6  The counting measure
 
Theoremcntmeas 23844 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 23845 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 23846 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( #  |`  ~P O )  =/= 
 vol
 
18.3.13.7  The Lebesgue measure - misc additions
 
Theoremvolss 23847 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 23848 The union of the Lebesgue measurable sets is  RR. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  U. dom  vol 
 =  RR
 
Theoremvoliune 23849 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 18962 and voliun 19009 (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 23850* The Lebesgue measure function is countably additive. This theorem is to volfiniun 19002 what voliune 23849 is to voliun 19009. (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 23851 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  vol  e.  (measures `  dom  vol )
 
18.3.13.8  The 'almost everywhere' relation
 
Syntaxcae 23852 Extend class notation to include the 'almost everywhere' relation.
 class a.e.
 
Syntaxcfae 23853 Extend class notation to include the 'almost everywhere' builder.
 class ~ a.e.
 
Definitiondf-ae 23854* 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 23855 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  Rel a.e.
 
Theorembrae 23856 '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 23857* '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 23858* 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 23859* 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 23860* 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 23861* 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 23862 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 23863* '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 ) )
 
18.3.13.9  Measurable functions
 
Syntaxcmbfm 23864 Extend class notation with the measurable functions builder.
 class MblFnM
 
Definitiondf-mbfm 23865* 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 19073 (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 23866* The predicate " F is a measurable function from the measurable space  S to the measurable space  T". Cf. ismbf 19083 (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 23867* 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 23868 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( ph  ->  F  e.  U. ran MblFnM )   =>    |-  ( ph  ->  Fun  F )
 
Theoremmbfmf 23869 A measurable function as a function with domain and codomain (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( SMblFnM T ) )   =>    |-  ( ph  ->  F : U. S --> U. T )
 
Theoremisanmbfm 23870 The predicate to be a measurable function (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( SMblFnM T ) )   =>    |-  ( ph  ->  F  e.  U. ran MblFnM )
 
Theoremmbfmcnvima 23871 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( SMblFnM T ) )   &    |-  ( ph  ->  A  e.  T )   =>    |-  ( ph  ->  ( `' F " A )  e.  S )
 
Theoremmbfmbfm 23872 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  F  e.  ( dom  MMblFnM (sigaGen `  J )
 ) )   =>    |-  ( ph  ->  F  e.  U. ran MblFnM )
 
Theoremmbfmcst 23873* A constant function is measurable. Cf. mbfconst 19088 (Contributed by Thierry Arnoux, 26-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  =  ( x  e.  U. S  |->  A ) )   &    |-  ( ph  ->  A  e.  U. T )   =>    |-  ( ph  ->  F  e.  ( SMblFnM T ) )
 
Theorem1stmbfm 23874 The first projection map is measurable with regard to the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( 1st  |`  ( U. S  X.  U. T ) )  e.  ( ( S ×s  T )MblFnM S ) )
 
Theorem2ndmbfm 23875 The second projection map is measurable with regard to the product sigma algebra (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( 2nd  |`  ( U. S  X.  U. T ) )  e.  ( ( S ×s  T )MblFnM T ) )
 
Theoremimambfm 23876* If the sigma-algebra in the range of a given function is generated by a collection of basic sets  K, then to check the measurability of that function, we need only consider inverse images of basic sets  a. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  =  (sigaGen `  K ) )   =>    |-  ( ph  ->  ( F  e.  ( SMblFnM T )  <->  ( F : U. S --> U. T  /\  A. a  e.  K  ( `' F " a )  e.  S ) ) )
 
Theoremcnmbfm 23877 A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  S  =  (sigaGen `  J ) )   &    |-  ( ph  ->  T  =  (sigaGen `  K ) )   =>    |-  ( ph  ->  F  e.  ( SMblFnM T ) )
 
Theoremmbfmco 23878 The composition of two measurable functions is measurable. ( cf. cnmpt11 17457) (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  R  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( RMblFnM S ) )   &    |-  ( ph  ->  G  e.  ( SMblFnM T ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  ( RMblFnM T ) )
 
Theoremmbfmco2 23879* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17459). (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  ( ph  ->  R  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( RMblFnM S ) )   &    |-  ( ph  ->  G  e.  ( RMblFnM T ) )   &    |-  H  =  ( x  e.  U. R  |->  <.
 ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ph  ->  H  e.  ( RMblFnM ( S ×s  T ) ) )
 
Theoremmbfmvolf 23880 Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  ( F  e.  ( dom  volMblFnM𝔅 )  ->  F : RR --> RR )
 
Theoremelmbfmvol2 23881 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.)
 |-  ( F  e.  ( dom  volMblFnM𝔅 )  ->  F  e. MblFn )
 
Theoremmbfmcnt 23882 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( O  e.  V  ->  ( ~P OMblFnM𝔅 )  =  ( RR  ^m  O ) )
 
18.3.13.10  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 23883* The base set for the generator of the Borel sigma algebra on  ( RR  X.  RR ) is indeed  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
 |->  ( x  X.  y
 ) )  =  ( RR  X.  RR )
 
Theoremdya2ub 23884 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  ( R  e.  RR+  ->  (
 1  /  ( 2 ^ ( |_ `  (
 1  -  ( 2logb
 R ) ) ) ) )  <  R )
 
Theoremsxbrsigalem0 23885* The closed half-spaces of  ( RR  X.  RR ) cover  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
 |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR )
 )  u.  ran  (
 f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) )  =  ( RR 
 X.  RR )
 
Theoremsxbrsigalem3 23886* The sigma-algebra generated by the closed half-spaces of  ( RR  X.  RR ) is a subset of the sigma-algebra generated by the closed sets of  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  u. 
 ran  ( f  e. 
 RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
 
Theoremdya2iocival 23887* The function  I returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19053. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  (
 2 ^ N ) ) [,) ( ( X  +  1 ) 
 /  ( 2 ^ N ) ) ) )
 
Theoremdya2iocress 23888* Dyadic intervals are subsets of  RR. (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  C_  RR )
 
Theoremdya2iocbrsiga 23889* Dyadic intervals are Borel sets of 
RR. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  e. 𝔅 )
 
Theoremdya2icobrsiga 23890* Dyadic intervals are Borel sets of 
RR. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |- 
 ran  I  C_ 𝔅
 
Theoremdya2icoseg 23891* For any point and any closed below, opened above interval of  RR 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.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  N  =  ( |_ `  ( 1  -  ( 2logb D ) ) )   =>    |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D )
 ) ) )
 
Theoremdya2icoseg2 23892* For any point and any opened interval of  RR 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.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E ) 
 ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
 
Theoremdya2iocrfn 23893* The function returning dyadic square covering for a given size has domain  ( ran  I  X.  ran  I ). (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  R  Fn  ( ran 
 I  X.  ran  I )
 
Theoremdya2iocct 23894* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |- 
 ran  R  ~<_  om
 
Theoremdya2iocnrect 23895* For any point of an opened rectangle in  ( RR  X.  RR ), 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.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   &    |-  B  =  ran  ( e  e.  ran  (,)
 ,  f  e.  ran  (,)  |->  ( e  X.  f
 ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
 
Theoremdya2iocnei 23896* For any point of an open set of the usual topology on  ( RR  X.  RR ) 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.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  ( ( A  e.  ( J  tX  J ) 
 /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
 
Theoremdya2iocuni 23897* Every open set of  ( RR  X.  RR ) is a union of closed below opened above dyadic rational rectangular subsets of  ( RR  X.  RR ). This union must be a countable union by dya2iocct 23894. (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  ( A  e.  ( J  tX  J )  ->  E. c  e.  ~P  ran 
 R U. c  =  A )
 
Theoremdya2iocucvr 23898* The dyadic rectangular set collection covers  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |- 
 U. ran  R  =  ( RR  X.  RR )
 
Theoremsxbrsigalem1 23899* The Borel algebra on  ( RR  X.  RR ) is a subset of the sigma algebra generated by the dyadic closed below, opened above rectangular subsets of  ( RR  X.  RR ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4 (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (sigaGen `  ran  R )
 
Theoremsxbrsigalem2 23900* The sigma-algebra generated by the dyadic closed below, opened above rectangular subsets of  ( RR  X.  RR ) is a subset of the sigma algebra generated by the closed half-spaces of  ( RR  X.  RR ). 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.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ran  R ) 
 C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR )
 )  u.  ran  (
 f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) ) )
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