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Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsge0p1 38101* The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  ( 0 [,] +oo ) )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  ( M
 ... ( N  +  1 ) )  |->  A ) )  =  ( (Σ^ `  ( k  e.  ( M ... N )  |->  A ) ) +e B ) )
 
Theoremsge0iunmptlemre 38102* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  (Σ^ `  ( k  e.  B  |->  C ) )  e. 
 RR )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  e.  RR* )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) )  e.  RR* )   &    |-  ( ph  ->  ( k  e.  U_ x  e.  A  B  |->  C ) :
 U_ x  e.  A  B
 --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  U_ x  e.  A  B  e.  _V )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) ) )
 
Theoremsge0fodjrnlem 38103* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned  0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C -onto-> A )   &    |-  ( ph  -> Disj  n  e.  C  ( F `  n ) )   &    |-  (
 ( ph  /\  n  e.  C )  ->  ( F `  n )  =  G )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  k  =  (/) )  ->  B  =  0 )   &    |-  Z  =  ( `' F " { (/) } )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  B ) )  =  (Σ^ `  ( n  e.  C  |->  D ) ) )
 
Theoremsge0fodjrn 38104* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned  0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C -onto-> A )   &    |-  ( ph  -> Disj  n  e.  C  ( F `  n ) )   &    |-  (
 ( ph  /\  n  e.  C )  ->  ( F `  n )  =  G )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  k  =  (/) )  ->  B  =  0 )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  =  (Σ^ `  ( n  e.  C  |->  D ) ) )
 
Theoremsge0iunmpt 38105* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) ) )
 
Theoremsge0iun 38106* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  X  =  U_ x  e.  A  B   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  ->  (Σ^ `  F )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( F  |`  B ) ) ) ) )
 
Theoremsge0nemnf 38107 The generalized sum of nonnegative extended reals is not minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =/= -oo )
 
Theoremsge0rpcpnf 38108* The sum of an infinite number of a positive constant, is +oo (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  -.  A  e.  Fin )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  = +oo )
 
Theoremsge0rernmpt 38109* If the sum of nonnegative extended reals is not +oo then no term is +oo. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  RR )   =>    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
 
Theoremsge0lefimpt 38110* A sum of nonnegative extended reals is smaller than a given extended real if and only if every finite subsum is smaller than it. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  C  e.  RR* )   =>    |-  ( ph  ->  (
 (Σ^ `  ( x  e.  A  |->  B ) )  <_  C 
 <-> 
 A. y  e.  ( ~P A  i^i  Fin )
 (Σ^ `  ( x  e.  y  |->  B ) )  <_  C ) )
 
Theoremnn0ssge0 38111 Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  NN0  C_  ( 0 [,) +oo )
 
Theoremsge0clmpt 38112* The generalized sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  ( 0 [,] +oo ) )
 
Theoremsge0ltfirpmpt2 38113* If the extended sum of nonnegative reals is not +oo, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  Y  e.  RR+ )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  RR )   =>    |-  ( ph  ->  E. y  e.  ( ~P A  i^i  Fin )
 (Σ^ `  ( x  e.  A  |->  B ) )  < 
 ( sum_ x  e.  y  B  +  Y )
 )
 
Theoremsge0isum 38114 If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F : Z --> ( 0 [,) +oo ) )   &    |-  G  =  seq M (  +  ,  F )   &    |-  ( ph  ->  G  ~~>  B )   =>    |-  ( ph  ->  (Σ^ `  F )  =  B )
 
Theoremsge0xrclmpt 38115* The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  RR* )
 
Theoremsge0xp 38116* Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ k ph   &    |-  ( z  = 
 <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  j  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 j  e.  A  |->  (Σ^ `  (
 k  e.  B  |->  C ) ) ) )  =  (Σ^ `  ( z  e.  ( A  X.  B )  |->  D ) ) )
 
Theoremsge0isummpt 38117* If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ k ph   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  seq
 M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  B )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  Z  |->  A ) )  =  B )
 
Theoremsge0ad2en 38118* The value of the infinite geometric series  2 ^ -u 1  +  2 ^ -u 2  +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  (
 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( n  e.  NN  |->  ( A  /  (
 2 ^ n ) ) ) )  =  A )
 
Theoremsge0isummpt2 38119* If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ k ph   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  seq
 M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  B )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  Z  |->  A ) )  = 
 sum_ k  e.  Z  A )
 
Theoremsge0xaddlem1 38120* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  U  C_  A )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  W  C_  A )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  <  ( sum_ k  e.  U  B  +  ( E  /  2
 ) ) )   &    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  C ) )  < 
 ( sum_ k  e.  W  C  +  ( E  /  2 ) ) )   &    |-  ( ph  ->  sup ( ran  ( x  e.  ( ~P A  i^i  Fin )  |-> 
 sum_ k  e.  x  ( B  +  C ) ) ,  RR* ,  <  )  e.  (
 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  B ) )  e. 
 RR )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  C ) )  e.  RR )   =>    |-  ( ph  ->  (
 (Σ^ `  ( k  e.  A  |->  B ) )  +  (Σ^ `  ( k  e.  A  |->  C ) ) ) 
 <_  ( sup ( ran  ( x  e.  ( ~P A  i^i  Fin )  |-> 
 sum_ k  e.  x  ( B  +  C ) ) ,  RR* ,  <  ) +e E ) )
 
Theoremsge0xaddlem2 38121* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  B ) )  e. 
 RR )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  C ) )  e.  RR )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  ( B +e C ) ) )  =  ( (Σ^ `  ( k  e.  A  |->  B ) ) +e (Σ^ `  ( k  e.  A  |->  C ) ) ) )
 
Theoremsge0xadd 38122* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  ( B +e C ) ) )  =  ( (Σ^ `  ( k  e.  A  |->  B ) ) +e (Σ^ `  ( k  e.  A  |->  C ) ) ) )
 
Theoremsge0fsummptf 38123* The generalized sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +oo (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  =  sum_ k  e.  A  B )
 
Theoremsge0snmptf 38124* A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  (
 0 [,] +oo ) )   &    |-  ( k  =  A  ->  B  =  C )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  { A }  |->  B ) )  =  C )
 
Theoremsge0ge0mpt 38125* The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  0  <_  (Σ^ `  ( k  e.  A  |->  B ) ) )
 
Theoremsge0repnfmpt 38126* The of nonnegative extended reals is a real number if and only if it is not +oo. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (
 (Σ^ `  ( k  e.  A  |->  B ) )  e. 
 RR 
 <->  -.  (Σ^ `  ( k  e.  A  |->  B ) )  = +oo ) )
 
Theoremsge0pnffigtmpt 38127* If the generalized sum of nonnegative reals is +oo, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  = +oo )   &    |-  ( ph  ->  Y  e.  RR )   =>    |-  ( ph  ->  E. x  e.  ( ~P A  i^i  Fin ) Y  <  (Σ^ `  (
 k  e.  x  |->  B ) ) )
 
Theoremsge0splitsn 38128* Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,] +oo ) )   &    |-  (
 k  =  B  ->  C  =  D )   &    |-  ( ph  ->  D  e.  (
 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  ( A  u.  { B }
 )  |->  C ) )  =  ( (Σ^ `  ( k  e.  A  |->  C ) ) +e D ) )
 
Theoremsge0pnffsumgt 38129* If the sum of nonnegative extended reals is +oo, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  = +oo )   &    |-  ( ph  ->  Y  e.  RR )   =>    |-  ( ph  ->  E. x  e.  ( ~P A  i^i  Fin ) Y  <  sum_ k  e.  x  B )
 
Theoremsge0gtfsumgt 38130* If the generalized sum of nonnegative reals is larger than a given number, then that number can be dominated by a finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  <  (Σ^ `  (
 k  e.  A  |->  B ) ) )   =>    |-  ( ph  ->  E. y  e.  ( ~P A  i^i  Fin ) C  <  sum_ k  e.  y  B )
 
Theoremsge0uzfsumgt 38131* If a real number is smaller than a generalized sum of nonnegative reals, then it is smaller than some finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  K )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  B  e.  ( 0 [,) +oo ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  <  (Σ^ `  (
 k  e.  Z  |->  B ) ) )   =>    |-  ( ph  ->  E. m  e.  Z  C  <  sum_ k  e.  ( K ... m ) B )
 
21.30.18.3  Measures

Proofs for most of the theorems in section 112 of [Fremlin1]

 
Syntaxcmea 38132 Extend class notation with the class of measures.
 class Meas
 
Definitiondf-mea 38133* Define the class of measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- Meas  =  { x  |  ( (
 ( x : dom  x --> ( 0 [,] +oo )  /\  dom  x  e. SAlg ) 
 /\  ( x `  (/) )  =  0 ) 
 /\  A. y  e.  ~P  dom 
 x ( ( y  ~<_ 
 om  /\ Disj  w  e.  y  w )  ->  ( x `
  U. y )  =  (Σ^ `  ( x  |`  y ) ) ) ) }
 
Theoremismea 38134* Express the predicate " M is a measure." Definition 112A of [Fremlin1] p. 14. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( M  e. Meas  <->  ( ( ( M : dom  M --> ( 0 [,] +oo )  /\  dom  M  e. SAlg ) 
 /\  ( M `  (/) )  =  0 ) 
 /\  A. x  e.  ~P  dom 
 M ( ( x  ~<_ 
 om  /\ Disj  y  e.  x  y )  ->  ( M `
  U. x )  =  (Σ^ `  ( M  |`  x ) ) ) ) )
 
Theoremdmmeasal 38135 The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   =>    |-  ( ph  ->  S  e. SAlg )
 
Theoremmeaf 38136 A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   =>    |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
 
Theoremmea0 38137 The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   =>    |-  ( ph  ->  ( M `  (/) )  =  0 )
 
Theoremnnfoctbdjlem 38138* There exists a mapping from  NN onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  C_  NN )   &    |-  ( ph  ->  G : A -1-1-onto-> X )   &    |-  ( ph  -> Disj  y  e.  X  y )   &    |-  F  =  ( n  e.  NN  |->  if ( ( n  =  1  \/  -.  ( n  -  1 )  e.  A ) ,  (/) ,  ( G `  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  E. f
 ( f : NN -onto->
 ( X  u.  { (/)
 } )  /\ Disj  n  e. 
 NN  ( f `  n ) ) )
 
Theoremnnfoctbdj 38139* There exists a mapping from  NN onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  ~<_  om )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  -> Disj  y  e.  X  y )   =>    |-  ( ph  ->  E. f
 ( f : NN -onto->
 ( X  u.  { (/)
 } )  /\ Disj  n  e. 
 NN  ( f `  n ) ) )
 
Theoremmeadjuni 38140* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  X 
 C_  S )   &    |-  ( ph  ->  X  ~<_  om )   &    |-  ( ph  -> Disj  x  e.  X  x )   =>    |-  ( ph  ->  ( M `  U. X )  =  (Σ^ `  ( M  |`  X ) ) )
 
Theoremmeacl 38141 The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  A  e.  S )   =>    |-  ( ph  ->  ( M `  A )  e.  ( 0 [,] +oo ) )
 
Theoremiundjiunlem 38142* The sets in the sequence  F are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  Z  =  ( ZZ>= `  N )   &    |-  F  =  ( n  e.  Z  |->  ( ( E `  n )  \  U_ i  e.  ( N..^ n ) ( E `  i
 ) ) )   &    |-  ( ph  ->  J  e.  Z )   &    |-  ( ph  ->  K  e.  Z )   &    |-  ( ph  ->  J  <  K )   =>    |-  ( ph  ->  ( ( F `  J )  i^i  ( F `  K ) )  =  (/) )
 
Theoremiundjiun 38143* Given a sequence  E of sets, a sequence  F of disjoint sets is built, such that the indexed union stays the same. As in the proof of Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ n ph   &    |-  Z  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  E : Z --> V )   &    |-  F  =  ( n  e.  Z  |->  ( ( E `  n )  \  U_ i  e.  ( N..^ n ) ( E `  i
 ) ) )   =>    |-  ( ph  ->  ( ( A. m  e.  Z  U_ n  e.  ( N ... m ) ( F `  n )  =  U_ n  e.  ( N ... m ) ( E `  n )  /\  U_ n  e.  Z  ( F `  n )  =  U_ n  e.  Z  ( E `  n ) )  /\ Disj  n  e.  Z  ( F `  n ) ) )
 
Theoremmeaxrcl 38144 The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  A  e.  S )   =>    |-  ( ph  ->  ( M `  A )  e.  RR* )
 
Theoremmeadjun 38145 The measure of the union of two disjoint sets is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( M `  ( A  u.  B ) )  =  ( ( M `  A ) +e
 ( M `  B ) ) )
 
Theoremmeassle 38146 The measure of a set is larger or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( M `  A )  <_  ( M `  B ) )
 
Theoremmeaunle 38147 The measure of the union of two sets is less or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   =>    |-  ( ph  ->  ( M `  ( A  u.  B ) )  <_  ( ( M `  A ) +e
 ( M `  B ) ) )
 
Theoremmeadjiunlem 38148* The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  G : X --> S )   &    |-  Y  =  {
 i  e.  X  |  ( G `  i )  =/=  (/) }   &    |-  ( ph  -> Disj  i  e.  X  ( G `  i ) )   =>    |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G )
 )  =  (Σ^ `  ( M  o.  G ) ) )
 
Theoremmeadjiun 38149* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  A  ~<_ 
 om )   &    |-  ( ph  -> Disj  k  e.  A  B )   =>    |-  ( ph  ->  ( M `  U_ k  e.  A  B )  =  (Σ^ `  ( k  e.  A  |->  ( M `  B ) ) ) )
 
Theoremismeannd 38150* Sufficient condition to prove that 
M is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  S  e. SAlg )   &    |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( M `  (/) )  =  0 )   &    |-  ( ( ph  /\  e : NN --> S  /\ Disj  n  e.  NN  ( e `  n ) )  ->  ( M `  U_ n  e.  NN  ( e `  n ) )  =  (Σ^ `  ( n  e.  NN  |->  ( M `  ( e `
  n ) ) ) ) )   =>    |-  ( ph  ->  M  e. Meas )
 
Theoremmeaiunlelem 38151* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ n ph   &    |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  Z  =  ( ZZ>= `  N )   &    |-  ( ph  ->  E : Z --> S )   &    |-  F  =  ( n  e.  Z  |->  ( ( E `
  n )  \  U_ i  e.  ( N..^ n ) ( E `
  i ) ) )   =>    |-  ( ph  ->  ( M `  U_ n  e.  Z  ( E `  n ) )  <_  (Σ^ `  ( n  e.  Z  |->  ( M `  ( E `
  n ) ) ) ) )
 
Theoremmeaiunle 38152* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ n ph   &    |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   &    |-  Z  =  ( ZZ>= `  N )   &    |-  ( ph  ->  E : Z --> S )   =>    |-  ( ph  ->  ( M ` 
 U_ n  e.  Z  ( E `  n ) )  <_  (Σ^ `  ( n  e.  Z  |->  ( M `  ( E `
  n ) ) ) ) )
 
Theorempsmeasurelem 38153*  M applied to a disjoint union of subsets of its domain is the sum of  M applied to such subset. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  H : X --> ( 0 [,] +oo ) )   &    |-  M  =  ( x  e.  ~P X  |->  (Σ^ `  ( H  |`  x ) ) )   &    |-  ( ph  ->  M : ~P X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  Y  C_  ~P X )   &    |-  ( ph  -> Disj  y  e.  Y  y )   =>    |-  ( ph  ->  ( M `  U. Y )  =  (Σ^ `  ( M  |`  Y ) ) )
 
Theorempsmeasure 38154* Point supported measure, Remark 112B (d) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  H : X --> ( 0 [,] +oo ) )   &    |-  M  =  ( x  e.  ~P X  |->  (Σ^ `  ( H  |`  x ) ) )   =>    |-  ( ph  ->  M  e. Meas )
 
21.30.18.4  Outer measures and Caratheodory's construction

Proofs for most of the theorems in section 113 of [Fremlin1]

 
Syntaxcome 38155 Extend class notation with the class of outer measures.
 class OutMeas
 
Definitiondf-ome 38156* Define the class of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- OutMeas  =  { x  |  ( (
 ( ( x : dom  x --> ( 0 [,] +oo )  /\  dom  x  =  ~P U. dom  x )  /\  ( x `  (/) )  =  0 ) 
 /\  A. y  e.  ~P  U.
 dom  x A. z  e.  ~P  y ( x `
  z )  <_  ( x `  y ) )  /\  A. y  e.  ~P  dom  x (
 y  ~<_  om  ->  ( x `
  U. y )  <_  (Σ^ `  ( x  |`  y ) ) ) ) }
 
Syntaxccaragen 38157 Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
 class CaraGen
 
Definitiondf-caragen 38158* Define the sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- CaraGen  =  ( o  e. OutMeas  |->  { e  e.  ~P U. dom  o  |  A. a  e.  ~P  U.
 dom  o ( ( o `  ( a  i^i  e ) ) +e ( o `
  ( a  \  e ) ) )  =  ( o `  a ) } )
 
Theoremcaragenval 38159* The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( O  e. OutMeas  ->  (CaraGen `  O )  =  { e  e.  ~P U. dom  O  |  A. a  e.  ~P  U.
 dom  O ( ( O `
  ( a  i^i  e ) ) +e ( O `  ( a  \  e ) ) )  =  ( O `  a ) } )
 
Theoremisome 38160* Express the predicate " O is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( O  e.  V  ->  ( O  e. OutMeas  <->  ( ( ( ( O : dom  O --> ( 0 [,] +oo )  /\  dom  O  =  ~P U. dom  O ) 
 /\  ( O `  (/) )  =  0 ) 
 /\  A. y  e.  ~P  U.
 dom  O A. z  e. 
 ~P  y ( O `
  z )  <_  ( O `  y ) )  /\  A. y  e.  ~P  dom  O (
 y  ~<_  om  ->  ( O `
  U. y )  <_  (Σ^ `  ( O  |`  y ) ) ) ) ) )
 
Theoremcaragenel 38161* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  ( E  e.  S 
 <->  ( E  e.  ~P U.
 dom  O  /\  A. a  e.  ~P  U. dom  O ( ( O `  ( a  i^i  E ) ) +e ( O `  ( a 
 \  E ) ) )  =  ( O `
  a ) ) ) )
 
Theoremomef 38162 An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   =>    |-  ( ph  ->  O : ~P X --> ( 0 [,] +oo ) )
 
Theoremome0 38163 The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   =>    |-  ( ph  ->  ( O `  (/) )  =  0 )
 
Theoremomessle 38164 The outer measure of a set is larger or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  B  C_  X )   &    |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( O `  A )  <_  ( O `  B ) )
 
Theoremomedm 38165 The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( O  e. OutMeas  ->  dom  O  =  ~P U. dom  O )
 
Theoremcaragensplit 38166 If  E is in the set generated by the Caratheodory's method, then it splits any set  A in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set  A. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  E  e.  S )   &    |-  ( ph  ->  A  C_  X )   =>    |-  ( ph  ->  (
 ( O `  ( A  i^i  E ) ) +e ( O `
  ( A  \  E ) ) )  =  ( O `  A ) )
 
Theoremcaragenelss 38167 An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  A  e.  S )   &    |-  X  =  U. dom  O   =>    |-  ( ph  ->  A  C_  X )
 
Theoremcarageneld 38168* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  E  e.  ~P X )   &    |-  ( ( ph  /\  a  e.  ~P X )  ->  ( ( O `
  ( a  i^i 
 E ) ) +e ( O `  ( a  \  E ) ) )  =  ( O `  a ) )   =>    |-  ( ph  ->  E  e.  S )
 
Theoremomecl 38169 The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   =>    |-  ( ph  ->  ( O `  A )  e.  ( 0 [,] +oo ) )
 
Theoremcaragenss 38170 The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  S  =  (CaraGen `  O )   =>    |-  ( O  e. OutMeas  ->  S  C_  dom 
 O )
 
Theoremomeunile 38171 The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  Y  C_  ~P X )   &    |-  ( ph  ->  Y  ~<_  om )   =>    |-  ( ph  ->  ( O `  U. Y ) 
 <_  (Σ^ `  ( O  |`  Y ) ) )
 
Theoremcaragen0 38172 The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  (/)  e.  S )
 
Theoremomexrcl 38173 The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   =>    |-  ( ph  ->  ( O `  A )  e.  RR* )
 
Theoremcaragenunidm 38174 The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  X  e.  S )
 
Theoremcaragensspw 38175 The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  S  C_  ~P X )
 
Theoremomessre 38176 If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( O `  A )  e. 
 RR )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  ( O `  B )  e.  RR )
 
Theoremcaragenuni 38177 The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  U. S  =  U. dom  O )
 
Theoremcaragenuncllem 38178 The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  E  e.  S )   &    |-  ( ph  ->  F  e.  S )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A 
 C_  X )   =>    |-  ( ph  ->  ( ( O `  ( A  i^i  ( E  u.  F ) ) ) +e ( O `
  ( A  \  ( E  u.  F ) ) ) )  =  ( O `  A ) )
 
Theoremcaragenuncl 38179 The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  E  e.  S )   &    |-  ( ph  ->  F  e.  S )   =>    |-  ( ph  ->  ( E  u.  F )  e.  S )
 
Theoremcaragendifcl 38180 The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  E  e.  S )   =>    |-  ( ph  ->  ( U. S  \  E )  e.  S )
 
Theoremcaragenfiiuncl 38181* The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   =>    |-  ( ph  ->  U_ k  e.  A  B  e.  S )
 
Theoremomeunle 38182 The outer measure of the union of two sets is less or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  B  C_  X )   =>    |-  ( ph  ->  ( O `  ( A  u.  B ) )  <_  ( ( O `  A ) +e
 ( O `  B ) ) )
 
Theoremomeiunle 38183* The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ n ph   &    |-  F/_ n E   &    |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  Z  =  ( ZZ>= `  N )   &    |-  ( ph  ->  E : Z --> ~P X )   =>    |-  ( ph  ->  ( O `  U_ n  e.  Z  ( E `  n ) )  <_  (Σ^ `  ( n  e.  Z  |->  ( O `  ( E `
  n ) ) ) ) )
 
Theoremomelesplit 38184 The outer measure of a set  A is less than or equal to the extended addition of the outer measures of the decomposition induced on  A by any  E. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   =>    |-  ( ph  ->  ( O `  A )  <_  ( ( O `  ( A  i^i  E ) ) +e ( O `  ( A 
 \  E ) ) ) )
 
Theoremomeiunltfirp 38185* If the outer measure of a countable union is not +oo, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  Z  =  ( ZZ>= `  N )   &    |-  ( ph  ->  E : Z --> ~P X )   &    |-  ( ph  ->  ( O `  U_ n  e.  Z  ( E `  n ) )  e. 
 RR )   &    |-  ( ph  ->  Y  e.  RR+ )   =>    |-  ( ph  ->  E. z  e.  ( ~P Z  i^i  Fin ) ( O `  U_ n  e.  Z  ( E `  n ) )  <  ( sum_ n  e.  z  ( O `
  ( E `  n ) )  +  Y ) )
 
Theoremomeiunlempt 38186* The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ n ph   &    |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  Z  =  (
 ZZ>= `  N )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  E  C_  X )   =>    |-  ( ph  ->  ( O `  U_ n  e.  Z  E )  <_  (Σ^ `  ( n  e.  Z  |->  ( O `  E ) ) ) )
 
Theoremcarageniuncllem1 38187* The outer measure of  A  i^i  ( G `  n ) is the sum of the outer measures of  A  i^i  ( F `  m ). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( O `  A )  e. 
 RR )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  E : Z --> S )   &    |-  G  =  ( n  e.  Z  |->  U_ i  e.  ( M ... n ) ( E `
  i ) )   &    |-  F  =  ( n  e.  Z  |->  ( ( E `
  n )  \  U_ i  e.  ( M..^ n ) ( E `
  i ) ) )   &    |-  ( ph  ->  K  e.  Z )   =>    |-  ( ph  ->  sum_
 n  e.  ( M
 ... K ) ( O `  ( A  i^i  ( F `  n ) ) )  =  ( O `  ( A  i^i  ( G `
  K ) ) ) )
 
Theoremcarageniuncllem2 38188* The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  X  =  U. dom  O   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( O `  A )  e. 
 RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  E : Z --> S )   &    |-  ( ph  ->  Y  e.  RR+ )   &    |-  G  =  ( n  e.  Z  |->  U_ i  e.  ( M ... n ) ( E `
  i ) )   &    |-  F  =  ( n  e.  Z  |->  ( ( E `
  n )  \  U_ i  e.  ( M..^ n ) ( E `
  i ) ) )   =>    |-  ( ph  ->  (
 ( O `  ( A  i^i  U_ n  e.  Z  ( E `  n ) ) ) +e
 ( O `  ( A  \  U_ n  e.  Z  ( E `  n ) ) ) )  <_  ( ( O `  A )  +  Y ) )
 
Theoremcarageniuncl 38189* The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  E : Z --> S )   =>    |-  ( ph  ->  U_ n  e.  Z  ( E `  n )  e.  S )
 
Theoremcaragenunicl 38190 The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  X  ~<_  om )   =>    |-  ( ph  ->  U. X  e.  S )
 
Theoremcaragensal 38191 Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  S  e. SAlg )
 
Theoremcaratheodorylem1 38192* Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  E : Z --> S )   &    |-  ( ph  -> Disj  n  e.  Z  ( E `  n ) )   &    |-  G  =  ( n  e.  Z  |->  U_ i  e.  ( M
 ... n ) ( E `  i ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  ( O `  ( G `  N ) )  =  (Σ^ `  ( n  e.  ( M ... N )  |->  ( O `  ( E `
  n ) ) ) ) )
 
Theoremcaratheodorylem2 38193* Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  X  =  U. dom  O   &    |-  S  =  (CaraGen `  O )   &    |-  ( ph  ->  E : NN --> S )   &    |-  ( ph  -> Disj  n  e.  NN  ( E `  n ) )   &    |-  G  =  ( k  e.  NN  |->  U_ n  e.  ( 1
 ... k ) ( E `  n ) )   =>    |-  ( ph  ->  ( O `  U_ n  e. 
 NN  ( E `  n ) )  =  (Σ^ `  ( n  e.  NN  |->  ( O `  ( E `
  n ) ) ) ) )
 
Theoremcaratheodory 38194 Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  O  e. OutMeas )   &    |-  S  =  (CaraGen `  O )   =>    |-  ( ph  ->  ( O  |`  S )  e. Meas )
 
Theorem0ome 38195* The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  O  =  ( x  e.  ~P X  |->  0 )   =>    |-  ( ph  ->  O  e. OutMeas )
 
Theoremisomenndlem 38196*  O is sub-additive w.r.t. countable indexed union, implies that  O is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  O : ~P X
 --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( O `  (/) )  =  0 )   &    |-  ( ph  ->  Y 
 C_  ~P X )   &    |-  (
 ( ph  /\  a : NN --> ~P X )  ->  ( O `  U_ n  e.  NN  ( a `  n ) )  <_  (Σ^ `  ( n  e.  NN  |->  ( O `  ( a `
  n ) ) ) ) )   &    |-  ( ph  ->  B  C_  NN )   &    |-  ( ph  ->  F : B -1-1-onto-> Y )   &    |-  A  =  ( n  e.  NN  |->  if ( n  e.  B ,  ( F `  n ) ,  (/) ) )   =>    |-  ( ph  ->  ( O ` 
 U. Y )  <_  (Σ^ `  ( O  |`  Y ) ) )
 
Theoremisomennd 38197* Sufficient condition to prove that 
O is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  O : ~P X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  ( O `  (/) )  =  0 )   &    |-  ( ( ph  /\  x  C_  X  /\  y  C_  x )  ->  ( O `
  y )  <_  ( O `  x ) )   &    |-  ( ( ph  /\  a : NN --> ~P X )  ->  ( O `  U_ n  e.  NN  (
 a `  n )
 )  <_  (Σ^ `  ( n  e.  NN  |->  ( O `  ( a `
  n ) ) ) ) )   =>    |-  ( ph  ->  O  e. OutMeas )
 
21.30.18.5  Lebesgue measure on n-dimensional Real numbers

Proof for the most important theorem in section 115 of [Fremlin1] (the plan is to complete the whole section with the next update)

 
Syntaxcovoln 38198 Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
 class voln*
 
Definitiondf-ovoln 38199* Define the outer measure for the space of multidimensional real numbers. The cardinality of 
x is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |- voln*  =  ( x  e. 
 Fin  |->  ( y  e. 
 ~P ( RR  ^m  x )  |->  if ( x  =  (/) ,  0 , inf ( { z  e.  RR*  |  E. i  e.  ( ( ( RR 
 X.  RR )  ^m  x )  ^m  NN ) ( y  C_  U_ j  e. 
 NN  X_ k  e.  x  ( ( [,)  o.  ( i `  j
 ) ) `  k
 )  /\  z  =  (Σ^ `  ( j  e.  NN  |->  prod_ k  e.  x  ( vol `  ( ( [,)  o.  ( i `  j ) ) `  k ) ) ) ) ) } ,  RR*
 ,  <  ) )
 ) )
 
Syntaxcvoln 38200 Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
 class voln
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