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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvolfiniune 29101* The Lebesgue measure function is countably additive. This theorem is to volfiniun 22548 what voliune 29100 is to voliun 22555. (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 29102 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  vol  e.  (measures `  dom  vol )
 
21.3.15.9  The Dirac delta measure
 
Syntaxcdde 29103 Extend class notation to include the Dirac delta measure.
 class δ
 
Definitiondf-dde 29104 Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
 |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
 
Theoremddeval1 29105 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
 |-  (
 ( A  C_  RR  /\  0  e.  A ) 
 ->  (δ `  A )  =  1 )
 
Theoremddeval0 29106 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
 |-  (
 ( A  C_  RR  /\ 
 -.  0  e.  A )  ->  (δ `  A )  =  0 )
 
Theoremddemeas 29107 The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
 |- δ  e.  (measures ` 
 ~P RR )
 
21.3.15.10  The 'almost everywhere' relation
 
Syntaxcae 29108 Extend class notation to include the 'almost everywhere' relation.
 class a.e.
 
Syntaxcfae 29109 Extend class notation to include the 'almost everywhere' builder.
 class ~ a.e.
 
Definitiondf-ae 29110* 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 29111 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
 |-  Rel a.e.
 
Theorembrae 29112 '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 29113* '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 29114* 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 29115* 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 29116* 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 29117* 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 29118 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 29119* '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 ) )
 
21.3.15.11  Measurable functions
 
Syntaxcmbfm 29120 Extend class notation with the measurable functions builder.
 class MblFnM
 
Definitiondf-mbfm 29121* 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-algebras  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 22625. (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 29122* The predicate " F is a measurable function from the measurable space  S to the measurable space  T". Cf. ismbf 22634. (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 29123* 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 29124 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( ph  ->  F  e.  U. ran MblFnM )   =>    |-  ( ph  ->  Fun  F )
 
Theoremmbfmf 29125 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 29126 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 29127 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 29128 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 29129* A constant function is measurable. Cf. mbfconst 22639. (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 29130 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 29131 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 29132* 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 29133 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 29134 The composition of two measurable functions is measurable. ( cf. cnmpt11 20726) (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 29135* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 20728). (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 29136 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 29137 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 29138 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 ) )
 
21.3.15.12  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 29139* 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 29140 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  ( R  e.  RR+  ->  (
 1  /  ( 2 ^ ( |_ `  (
 1  -  ( 2 logb  R ) ) ) ) )  <  R )
 
Theoremsxbrsigalem0 29141* 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 29142* 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 29143* The function  I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 22606. (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 29144* 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 29145* 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 29146* 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 29147* For any point and any closed-below, open-above interval of  RR centered on that point, there is a closed-below open-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  -  ( 2 logb  D ) ) )   =>    |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D ) ) ) )
 
Theoremdya2icoseg2 29148* For any point and any open interval of  RR containing that point, there is a closed-below open-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 29149* 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 29150* 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 29151* For any point of an open rectangle in 
( RR  X.  RR ), there is a closed-below open-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 29152* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is a closed-below open-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 29153* Every open set of  ( RR  X.  RR ) is a union of closed-below open-above dyadic rational rectangular subsets of  ( RR  X.  RR ). This union must be a countable union by dya2iocct 29150. (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 29154* 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 29155* The Borel algebra on  ( RR  X.  RR ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-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 29156* The sigma-algebra generated by the dyadic closed-below, open-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 ) ) ) ) )
 
Theoremsxbrsigalem4 29157* The Borel algebra on  ( RR  X.  RR ) is generated by the dyadic closed-below, open-above rectangular subsets of  ( RR  X.  RR ). Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed-below, open-above instead of open-below, closed-above in the proof as they are ultimately generated by the floor function. (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
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ran  R )
 
Theoremsxbrsigalem5 29158* First direction for sxbrsiga 29160. (Contributed by Thierry Arnoux, 22-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
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (𝔅 ×s 𝔅 )
 
Theoremsxbrsigalem6 29159 First direction for sxbrsiga 29160, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (𝔅 ×s 𝔅 )
 
Theoremsxbrsiga 29160 The product sigma-algebra  (𝔅 ×s 𝔅 ) is the Borel algebra on  ( RR  X.  RR ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )
 
21.3.15.13  Caratheodory's extension theorem

In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure  R. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 29173) - it is monotone (omsmon 29174) - it is countably sub-additive (omssubadd 29176) See Definition 1.11.1 of [Bogachev] p. 41.

 
Syntaxcoms 29161 Class declaration for the outer measure construction function.
 class toOMeas
 
Syntaxcomsold 29162 Class declaration for the outer measure construction function (old version).
 class toOMeas
 
Definitiondf-oms 29163* Define a function constructing an outer measure. See omsval 29165 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |- toOMeas  =  ( r  e.  _V  |->  ( a  e.  ~P U. dom  r  |-> inf ( ran  ( x  e.  { z  e.  ~P dom  r  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
 y  e.  x ( r `  y ) ) ,  ( 0 [,] +oo ) ,  <  ) ) )
 
Definitiondf-omsOLD 29164* Define a function constructing an outer measure. See omsval 29165 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of df-oms 29163 as of 4-Oct-2020. (New usage is discouraged.)
 |- toOMeas  =  ( r  e.  _V  |->  ( a  e.  ~P U. dom  r  |->  sup ( ran  ( x  e.  { z  e.  ~P dom  r  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
 y  e.  x ( r `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
 
Theoremomsval 29165* Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  ( R  e.  _V  ->  (toOMeas `  R )  =  ( a  e.  ~P U. dom  R  |-> inf ( ran  ( x  e.  { z  e.  ~P dom  R  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
 y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  <  ) ) )
 
Theoremomsfval 29166* Value of the outer measure evaluated for a given set  A. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_ 
 U. Q )  ->  ( (toOMeas `  R ) `  A )  = inf ( ran  ( x  e.  {
 z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  <  ) )
 
Theoremomscl 29167* A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  e.  ~P U. dom  R )  ->  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A 
 C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) )  C_  ( 0 [,] +oo ) )
 
Theoremomsf 29168 A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo ) )  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
 
TheoremomsvalOLD 29169* Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsval 29165 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( R  e.  _V  ->  (toOMeas `  R )  =  ( a  e.  ~P U. dom  R  |->  sup ( ran  ( x  e.  { z  e.  ~P dom  R  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
 y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
 
TheoremomsfvalOLD 29170* Value of the outer measure evaluated for a given set  A. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsfval 29166 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_ 
 U. Q )  ->  ( (toOMeas `  R ) `  A )  =  sup ( ran  ( x  e. 
 { z  e.  ~P dom  R  |  ( A 
 C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
 
TheoremomsclOLD 29171* A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) Obsolete version of omscl 29167 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  e.  ~P U. dom  R )  ->  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A 
 C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) )  C_  ( 0 [,] +oo ) )
 
TheoremomsfOLD 29172 A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) Obsolete version of omsf 29168 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo ) )  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
 
Theoremoms0 29173 A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (/)  e. 
 dom  R )   &    |-  ( ph  ->  ( R `  (/) )  =  0 )   =>    |-  ( ph  ->  ( M `  (/) )  =  0 )
 
Theoremomsmon 29174 A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A 
 C_  B )   &    |-  ( ph  ->  B  C_  U. Q )   =>    |-  ( ph  ->  ( M `  A )  <_  ( M `  B ) )
 
Theoremomssubaddlem 29175* For any small margin  E, we can find a covering approaching the outer measure of a set  A by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A 
 C_  U. Q )   &    |-  ( ph  ->  ( M `  A )  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
 C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
  A )  +  E ) )
 
Theoremomssubadd 29176* A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  y  e.  X ) 
 ->  A  C_  U. Q )   &    |-  ( ph  ->  X  ~<_  om )   =>    |-  ( ph  ->  ( M `  U_ y  e.  X  A )  <_ Σ* y  e.  X ( M `  A ) )
 
Theoremoms0OLD 29177 A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of oms0 29173 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (/)  e. 
 dom  R )   &    |-  ( ph  ->  ( R `  (/) )  =  0 )   =>    |-  ( ph  ->  ( M `  (/) )  =  0 )
 
TheoremomsmonOLD 29178 A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsmon 29174 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A 
 C_  B )   &    |-  ( ph  ->  B  C_  U. Q )   =>    |-  ( ph  ->  ( M `  A )  <_  ( M `  B ) )
 
TheoremomssubaddlemOLD 29179* For any small margin  E, we can find a covering approaching the outer measure of a set  A by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) Obsolete version of omssubaddlem 29175 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A 
 C_  U. Q )   &    |-  ( ph  ->  ( M `  A )  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
 C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
  A )  +  E ) )
 
TheoremomssubaddOLD 29180* A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) Obsolete version of omssubadd 29176 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  (toOMeas `  R )   &    |-  ( ph  ->  Q  e.  V )   &    |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  y  e.  X ) 
 ->  A  C_  U. Q )   &    |-  ( ph  ->  X  ~<_  om )   =>    |-  ( ph  ->  ( M `  U_ y  e.  X  A )  <_ Σ* y  e.  X ( M `  A ) )
 
Syntaxccarsg 29181 Class declaration for the Caratheodory sigma-Algebra construction.
 class toCaraSiga
 
Definitiondf-carsg 29182* Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 29183 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
 |- toCaraSiga  =  ( m  e.  _V  |->  { a  e.  ~P U. dom  m  |  A. e  e.  ~P  U. dom  m ( ( m `  ( e  i^i  a ) ) +e ( m `  ( e 
 \  a ) ) )  =  ( m `
  e ) }
 )
 
Theoremcarsgval 29183* Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (toCaraSiga `  M )  =  {
 a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `
  ( e  i^i  a ) ) +e ( M `  ( e  \  a ) ) )  =  ( M `  e ) } )
 
Theoremcarsgcl 29184 Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (toCaraSiga `  M )  C_  ~P O )
 
Theoremelcarsg 29185* Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  ( A  e.  (toCaraSiga `  M ) 
 <->  ( A  C_  O  /\  A. e  e.  ~P  O ( ( M `
  ( e  i^i 
 A ) ) +e ( M `  ( e  \  A ) ) )  =  ( M `  e ) ) ) )
 
Theorembaselcarsg 29186 The universe set,  O, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   =>    |-  ( ph  ->  O  e.  (toCaraSiga `
  M ) )
 
Theorem0elcarsg 29187 The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   =>    |-  ( ph  ->  (/)  e.  (toCaraSiga `  M ) )
 
Theoremcarsguni 29188 The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   =>    |-  ( ph  ->  U. (toCaraSiga `  M )  =  O )
 
Theoremelcarsgss 29189 Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  A 
 C_  O )
 
Theoremdifelcarsg 29190 The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  ( O  \  A )  e.  (toCaraSiga `  M ) )
 
Theoreminelcarsg 29191* The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  (toCaraSiga `  M ) )   &    |-  (
 ( ph  /\  a  e. 
 ~P O  /\  b  e.  ~P O )  ->  ( M `  ( a  u.  b ) ) 
 <_  ( ( M `  a ) +e
 ( M `  b
 ) ) )   &    |-  ( ph  ->  B  e.  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  ( A  i^i  B )  e.  (toCaraSiga `  M ) )
 
Theoremunelcarsg 29192* The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  (toCaraSiga `  M ) )   &    |-  (
 ( ph  /\  a  e. 
 ~P O  /\  b  e.  ~P O )  ->  ( M `  ( a  u.  b ) ) 
 <_  ( ( M `  a ) +e
 ( M `  b
 ) ) )   &    |-  ( ph  ->  B  e.  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  ( A  u.  B )  e.  (toCaraSiga `  M ) )
 
Theoremdifelcarsg2 29193* The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  (toCaraSiga `  M ) )   &    |-  (
 ( ph  /\  a  e. 
 ~P O  /\  b  e.  ~P O )  ->  ( M `  ( a  u.  b ) ) 
 <_  ( ( M `  a ) +e
 ( M `  b
 ) ) )   &    |-  ( ph  ->  B  e.  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  ( A  \  B )  e.  (toCaraSiga `  M ) )
 
Theoremcarsgmon 29194* Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  ~P O )   &    |-  (
 ( ph  /\  x  C_  y  /\  y  e.  ~P O )  ->  ( M `
  x )  <_  ( M `  y ) )   =>    |-  ( ph  ->  ( M `  A )  <_  ( M `  B ) )
 
Theoremcarsgsigalem 29195* Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   =>    |-  ( ( ph  /\  e  e.  ~P O  /\  f  e.  ~P O )  ->  ( M `  ( e  u.  f ) ) 
 <_  ( ( M `  e ) +e
 ( M `  f
 ) ) )
 
Theoremfiunelcarsg 29196* The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  (toCaraSiga `  M ) )   =>    |-  ( ph  ->  U. A  e.  (toCaraSiga `  M ) )
 
Theoremcarsgclctunlem1 29197* Lemma for carsgclctun 29201. (Contributed by Thierry Arnoux, 23-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  (toCaraSiga `  M ) )   &    |-  ( ph  -> Disj  y  e.  A  y )   &    |-  ( ph  ->  E  e.  ~P O )   =>    |-  ( ph  ->  ( M `  ( E  i^i  U. A ) )  = Σ* y  e.  A ( M `
  ( E  i^i  y ) ) )
 
Theoremcarsggect 29198* The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   &    |-  ( ph  ->  -.  (/)  e.  A )   &    |-  ( ph  ->  A  ~<_  om )   &    |-  ( ph  ->  A  C_  (toCaraSiga `  M ) )   &    |-  ( ph  -> Disj  y  e.  A  y )   &    |-  ( ( ph  /\  x  C_  y  /\  y  e.  ~P O )  ->  ( M `  x )  <_  ( M `
  y ) )   =>    |-  ( ph  -> Σ* z  e.  A ( M `  z ) 
 <_  ( M `  U. A ) )
 
Theoremcarsgclctunlem2 29199* Lemma for carsgclctun 29201. (Contributed by Thierry Arnoux, 25-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   &    |-  ( ( ph  /\  x  C_  y  /\  y  e.  ~P O )  ->  ( M `  x )  <_  ( M `
  y ) )   &    |-  ( ph  -> Disj  k  e.  NN  A )   &    |-  ( ( ph  /\  k  e.  NN )  ->  A  e.  (toCaraSiga `  M ) )   &    |-  ( ph  ->  E  e.  ~P O )   &    |-  ( ph  ->  ( M `  E )  =/= +oo )   =>    |-  ( ph  ->  (
 ( M `  ( E  i^i  U_ k  e.  NN  A ) ) +e ( M `  ( E  \  U_ k  e.  NN  A ) ) )  <_  ( M `  E ) )
 
Theoremcarsgclctunlem3 29200* Lemma for carsgclctun 29201. (Contributed by Thierry Arnoux, 24-May-2020.)
 |-  ( ph  ->  O  e.  V )   &    |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  ~<_  om 
 /\  x  C_  ~P O )  ->  ( M `  U. x )  <_ Σ* y  e.  x ( M `  y ) )   &    |-  ( ( ph  /\  x  C_  y  /\  y  e.  ~P O )  ->  ( M `  x )  <_  ( M `
  y ) )   &    |-  ( ph  ->  A  ~<_  om )   &    |-  ( ph  ->  A  C_  (toCaraSiga `  M ) )   &    |-  ( ph  ->  E  e.  ~P O )   =>    |-  ( ph  ->  (
 ( M `  ( E  i^i  U. A ) ) +e ( M `
  ( E  \  U. A ) ) ) 
 <_  ( M `  E ) )
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