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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsigagenval 23501* 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 23502 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 23503 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 23504 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 23505 A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  dom sigaGen  =  _V
 
Theoremsssigagen 23506 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 23507 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 23508 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 23509 A 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 23510 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 )
 
Theoremsigagenid 23511 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 )
 
18.3.34  The Borel Algebra on real numbers
 
Syntaxcbrsiga 23512 The Borel Algebra on real numbers, usually a gothic B
 class 𝔅
 
Definitiondf-brsiga 23513 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 23514 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigaGen " Top )
 
Theorembrsigarn 23515 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |- 𝔅  e.  (sigAlgebra `  RR )
 
Theorembrsigasspwrn 23516 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |- 𝔅 
 C_  ~P RR
 
Theoremunibrsiga 23517 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  U.𝔅  =  RR
 
Theoremcldssbrsiga 23518 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 ) )
 
18.3.35  Product Sigma-Algebra
 
Syntaxcsx 23519 Extend class notation with the product sigma-algebra operation.
 class ×s
 
Definitiondf-sx 23520* Define the product sigma-algebra operation, analogue to df-tx 17257. (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 23521* 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 23522 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 23523 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 23524 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 23525 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 ) )
 
18.3.36  Measures
 
Syntaxcmeas 23526 Extend class notation to include the class of measures.
 class measures
 
Definitiondf-meas 23527* 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 23528 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 23529* 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 23530* 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 23531* 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 23532 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 23533 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 23534 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 23535 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 23536 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  ( M  e.  (measures `  S )  ->  ( M `  (/) )  =  0 )
 
Theoremmeasvun 23537* 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 23538 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 ) ) ) )
 
Theoremmeasxun 23539 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 23540* Lemma for measvuni 23542 (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 23541* Lemma for measvuni 23542. (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 23542* 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 23543 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 ) )
 
Theoremmeasiuns 23544* 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 23545 and meascnbl 23546 (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 23545* 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 23546* A measure is continuous from below. Cf. volsup 18913. (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 23547* Lemma for measinb 23548 (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 23548* 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 23549 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 23550* 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 23551* 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 23552 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 )
 )
 
Theoremcntmeas 23553 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 23554 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 ) )
 
18.3.37  Measurable functions
 
Syntaxcmbfm 23555 Extend class notation with the measurable functions builder.
 class MblFnM
 
Definitiondf-mbfm 23556* 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 18975 (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 23557* The predicate " F is a measurable function from the measurable space  S to the measurable space  T". Cf. ismbf 18985 (Contributed by Thierry Arnoux, 23-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( F  e.  (MblFnM `  <. S ,  T >. )  <->  ( F  e.  ( U. T  ^m  U. S )  /\  A. x  e.  T  ( `' F " x )  e.  S ) ) )
 
Theoremelunirnmbfm 23558* 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 23559 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( ph  ->  F  e.  U. ran MblFnM )   =>    |-  ( ph  ->  Fun  F )
 
Theoremmbfmf 23560 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.  (MblFnM ` 
 <. S ,  T >. ) )   =>    |-  ( ph  ->  F : U. S --> U. T )
 
Theoremisanmbfm 23561 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.  (MblFnM ` 
 <. S ,  T >. ) )   =>    |-  ( ph  ->  F  e.  U. ran MblFnM )
 
Theoremmbfmcnvima 23562 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.  (MblFnM ` 
 <. S ,  T >. ) )   &    |-  ( ph  ->  A  e.  T )   =>    |-  ( ph  ->  ( `' F " A )  e.  S )
 
Theoremmbfmbfm 23563 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.  (MblFnM ` 
 <. dom  M ,  (sigaGen `  J ) >. ) )   =>    |-  ( ph  ->  F  e.  U.
 ran MblFnM )
 
Theoremmbfmcst 23564* A constant function is measurable. Cf. mbfconst 18990 (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.  (MblFnM `  <. S ,  T >. ) )
 
Theorem1stmbfm 23565 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.  (MblFnM `  <. ( S ×s  T ) ,  S >. ) )
 
Theorem2ndmbfm 23566 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.  (MblFnM `  <. ( S ×s  T ) ,  T >. ) )
 
Theoremimambfm 23567* 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.  (MblFnM `  <. S ,  T >. )  <->  ( F : U. S --> U. T  /\  A. a  e.  K  ( `' F " a )  e.  S ) ) )
 
Theoremcnmbfm 23568 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.  (MblFnM `  <. S ,  T >. ) )
 
Theoremmbfmco 23569 The composition of two measurable functions is measurable. ( cf. cnmpt11 17357) (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.  (MblFnM `  <. R ,  S >. ) )   &    |-  ( ph  ->  G  e.  (MblFnM ` 
 <. S ,  T >. ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  (MblFnM `  <. R ,  T >. ) )
 
Theoremmbfmco2 23570* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17359). (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.  (MblFnM `  <. R ,  S >. ) )   &    |-  ( ph  ->  G  e.  (MblFnM ` 
 <. R ,  T >. ) )   &    |-  H  =  ( x  e.  U. R  |->  <.
 ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ph  ->  H  e.  (MblFnM `  <. R ,  ( S ×s  T ) >. ) )
 
Theoremmbfmvolf 23571 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.  (MblFnM `  <. dom  vol , 𝔅 >. ) 
 ->  F : RR --> RR )
 
Theoremelmbfmvol2 23572 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.  (MblFnM `  <. dom  vol , 𝔅 >. ) 
 ->  F  e. MblFn )
 
Theoremmbfmcnt 23573 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( O  e.  V  ->  (MblFnM `  <. ~P O , 𝔅 >. )  =  ( RR  ^m  O ) )
 
18.3.38  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 23574* 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 23575 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  ( R  e.  RR+  ->  (
 1  /  ( 2 ^ ( |_ `  (
 1  -  ( 2logb
 R ) ) ) ) )  <  R )
 
Theoremdya2iocival 23576* The function  I returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 18955. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) (
 ( X  +  1 )  /  ( 2 ^ N ) ) ) )
 
Theoremdya2iocress 23577* 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 ) ) ) )   &    |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  C_  RR )
 
Theoremdya2iocbrsiga 23578* 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 ) ) ) )   &    |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  e. 𝔅 )
 
Theoremdya2iocseg 23579* 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 ) ) ) )   &    |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   &    |-  N  =  ( |_ `  (
 1  -  ( 2logb
 D ) ) )   &    |-  G  =  ( I  o.  `' ( 1st  |`  ( _V 
 X.  { N } )
 ) )   =>    |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ( G " ZZ ) ( X  e.  b  /\  b  C_  ( ( X  -  D ) [,) ( X  +  D ) ) ) )
 
Theoremdya2iocrfn 23580* The function returning dyadic square covering for a given size has domain  ZZ. (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  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  R  Fn  ZZ
 
Theoremdya2iocct 23581* The dyadic rectangle set is countable. (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  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  U. ran  R  ~<_  om
 
Theoremdya2iocrrnval 23582* The value of a dyadic square cover of 
( RR  X.  RR ). (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
 y I n ) ) ) )   =>    |-  ( N  e.  ZZ  ->  ( V  e.  ( R `  N )  <->  E. x  e.  ZZ  E. y  e.  ZZ  V  =  ( ( x I N )  X.  (
 y I N ) ) ) )
 
18.3.39  Integration with respect to a Measure
 
Syntaxcibfm 23583 Extend class notation with the integrable functions builder.
 class IblFnM
 
Syntaxcitgm 23584 Extend class notation with the Measure integral.
 class𝜇 A F  _d M
 
Definitiondf-itgmTMP 23585* Define the measure integral, for functions from a measurable space to  ( 0 [,]  +oo ). Unlike the definition of the Lebesgue integral, df-itg 18979, the function to be integrated is given as a function and not as a class with a free variable  x. In this definition,  M is the measure to be used to integrate  F.

The current definition falls back to the two cases used in propability theory: - the Lebesgue measure, used for continuous distributions - the Counting measure, used for discrete distributions

For other cases, it leads to  _V, which results in other integrals not to be defined (not sets).

This is currently a placeholder, the long term goal being to construct the measure integral for all measures - this would go similarly as was done for the Lebesgue integral. Therefore this definition and the following theorems are labelled "TMP", to show they are based on a temporary construct and will need re-work.

(Contributed by Thierry Arnoux, 16-Jan-2017.)

 |-𝜇 A F  _d M  =  if ( M  =  vol ,  S. A ( F `  x )  _d x ,  if ( E. o  M  =  ( #  |`  ~P o
 ) , Σ* x  e.  A ( F `  x ) ,  _V ) )
 
Definitiondf-ibfm 23586* Define the set of integrable functions with respect to a measure  m. (Contributed by Thierry Arnoux, 16-Jan-2017.)
 |- IblFnM  =  ( m  e.  U. ran measures  |->  { f  e.  (MblFnM `  <. dom  m , 𝔅 >.
 )  |  (𝜇 U. dom  m  f  _d m  e.  RR  /\  ran  f  C_  ( 0 [,)  +oo ) ) }
 )
 
TheoremitgmvolTMP 23587* When the Lebesgue integral is used, the measure integral is the Lebesgue integral. This is currently a placeholder and a complete (and complex) proof will be necessary once the measure integral will be properly defined. (Contributed by Thierry Arnoux, 26-Jan-2017.)
 |-𝜇 A F  _d vol  =  S. A ( F `
  x )  _d x
 
TheoremitgmcntTMP 23588* When the counting measure is used, the measure integral is the usual sum operation. This is currently a placeholder and a complete proof will be necessary once the measure integral will be properly defined. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( O  e.  V  ->𝜇 A F  _d ( #  |`  ~P O )  = Σ* x  e.  A ( F `  x ) )
 
Theoremitgmeq123dTMP 23589 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  M  =  P )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 B G  _d P )
 
Theoremitgmeq1dTMP 23590 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 B F  _d M )
 
Theoremitgmeq2dTMP 23591 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 A G  _d M )
 
Theoremitgmeq3dTMP 23592 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  M  =  P )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 A F  _d P )
 
Theoremisibfm 23593 The property of being integrable with respect to a given measure  M. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  ( M  e.  U. ran measures  ->  ( F  e.  (IblFnM `  M ) 
 <->  ( F  e.  (MblFnM ` 
 <. dom  M , 𝔅
 >. )  /\  ran  F  C_  ( 0 [,)  +oo )  /\𝜇 U. dom  M  F  _d M  e.  RR )
 ) )
 
18.3.40  Indicator Functions
 
Syntaxcind 23594 Extend class notation with the indicator function generator.
 class D7ED;
 
Definitiondf-ind 23595* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
 |- D7ED;  =  ( o  e.  _V  |->  ( a  e.  ~P o  |->  ( x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindv 23596* Value of the indicator function generator with domain  O. (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  ( O  e.  V  ->  (D7ED; `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindval 23597* Value of the indicator function generator for a set  A and a domain  O. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (D7ED; `  O ) `  A )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
 
Theoremindval2 23598 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (D7ED; `  O ) `  A )  =  ( ( A  X.  { 1 } )  u.  ( ( O  \  A )  X.  { 0 } ) ) )
 
Theoremindf 23599 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (D7ED; `  O ) `  A ) : O --> { 0 ,  1 } )
 
Theoremindfval 23600 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  ( ( (D7ED; `  O ) `  A ) `  X )  =  if ( X  e.  A ,  1 ,  0 ) )
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