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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovolicopnf 22101 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo )
 )  = +oo )
 
Theoremovolre 22102 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( vol* `  RR )  = +oo
 
Theoremismbl 22103* The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set  x in a "nice" way, that is, if the measure of the pieces  x  i^i  A and  x  \  A sum up to the measure of 
x (assuming that the measure of 
x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) ) ) )
 
Theoremismbl2 22104* From ovolun 22076, it suffices to show that the measure of  x is at least the sum of the measures of  x  i^i  A and  x  \  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) 
 <_  ( vol* `  x ) ) ) )
 
Theoremvolres 22105 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol  =  ( vol*  |`  dom  vol )
 
Theoremvolf 22106 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol : dom  vol --> ( 0 [,] +oo )
 
Theoremmblvol 22107 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( vol `  A )  =  ( vol* `
  A ) )
 
Theoremmblss 22108 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  A  C_  RR )
 
Theoremmblsplit 22109 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  C_  RR  /\  ( vol* `  B )  e.  RR )  ->  ( vol* `  B )  =  ( ( vol* `  ( B  i^i  A ) )  +  ( vol* `  ( B  \  A ) ) ) )
 
Theoremvolss 22110 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol  /\  A  C_  B )  ->  ( vol `  A )  <_  ( vol `  B ) )
 
Theoremcmmbl 22111 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( RR  \  A )  e.  dom  vol )
 
Theoremnulmbl 22112 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  e.  dom 
 vol )
 
Theoremnulmbl2 22113* A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has  vol* ( A )  =  0 while "outer measure zero" means that for any  x there is a  y containing  A with volume less than  x. Assuming AC, these notions are equivalent (because the intersection of all such  y is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A. x  e.  RR+  E. y  e.  dom  vol ( A  C_  y  /\  ( vol* `  y )  <_  x ) 
 ->  A  e.  dom  vol )
 
Theoremunmbl 22114 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  u.  B )  e.  dom  vol )
 
Theoremshftmbl 22115* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR )  ->  { x  e. 
 RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
 
Theorem0mbl 22116 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  (/)  e.  dom  vol
 
Theoremrembl 22117 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |- 
 RR  e.  dom  vol
 
Theoreminmbl 22118 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  i^i  B )  e.  dom  vol )
 
Theoremdifmbl 22119 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  \  B )  e.  dom  vol )
 
Theoremfiniunmbl 22120* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  B  e.  dom  vol )  -> 
 U_ k  e.  A  B  e.  dom  vol )
 
Theoremvolun 22121 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( ( A  e.  dom  vol  /\  B  e.  dom  vol  /\  ( A  i^i  B )  =  (/) )  /\  ( ( vol `  A )  e.  RR  /\  ( vol `  B )  e.  RR ) )  ->  ( vol `  ( A  u.  B ) )  =  (
 ( vol `  A )  +  ( vol `  B ) ) )
 
Theoremvolinun 22122 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( ( A  e.  dom  vol  /\  B  e.  dom  vol )  /\  (
 ( vol `  A )  e.  RR  /\  ( vol `  B )  e.  RR ) )  ->  ( ( vol `  A )  +  ( vol `  B ) )  =  (
 ( vol `  ( A  i^i  B ) )  +  ( vol `  ( A  u.  B ) ) ) )
 
Theoremvolfiniun 22123* The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  ( B  e.  dom  vol  /\  ( vol `  B )  e.  RR )  /\ Disj  k  e.  A  B )  ->  ( vol `  U_ k  e.  A  B )  = 
 sum_ k  e.  A  ( vol `  B )
 )
 
Theoremiundisj 22124* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  NN  A  =  U_ n  e.  NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisj2 22125* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremvoliunlem1 22126* Lemma for voliun 22130. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN  ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol* `  ( E  i^i  ( F `  n ) ) ) )   &    |-  ( ph  ->  E 
 C_  RR )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   =>    |-  (
 ( ph  /\  k  e. 
 NN )  ->  (
 (  seq 1 (  +  ,  H ) `  k
 )  +  ( vol* `  ( E  \ 
 U. ran  F )
 ) )  <_  ( vol* `  E ) )
 
Theoremvoliunlem2 22127* Lemma for voliun 22130. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN  ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol* `  ( x  i^i  ( F `  n ) ) ) )   =>    |-  ( ph  ->  U. ran  F  e.  dom  vol )
 
Theoremvoliunlem3 22128* Lemma for voliun 22130. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN  ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol* `  ( x  i^i  ( F `  n ) ) ) )   &    |-  S  =  seq 1 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  ( F `  n ) ) )   &    |-  ( ph  ->  A. i  e.  NN  ( vol `  ( F `  i ) )  e. 
 RR )   =>    |-  ( ph  ->  ( vol `  U. ran  F )  =  sup ( ran 
 S ,  RR* ,  <  ) )
 
Theoremiunmbl 22129 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( A. n  e. 
 NN  A  e.  dom  vol 
 ->  U_ n  e.  NN  A  e.  dom  vol )
 
Theoremvoliun 22130 The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  S  =  seq 1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  A ) )   =>    |-  ( ( A. n  e.  NN  ( A  e.  dom  vol  /\  ( vol `  A )  e.  RR )  /\ Disj  n  e. 
 NN  A )  ->  ( vol `  U_ n  e. 
 NN  A )  = 
 sup ( ran  S ,  RR* ,  <  )
 )
 
Theoremvolsuplem 22131* Lemma for volsup 22132. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( ( A. n  e.  NN  ( F `  n )  C_  ( F `
  ( n  +  1 ) )  /\  ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) ) ) 
 ->  ( F `  A )  C_  ( F `  B ) )
 
Theoremvolsup 22132* The volume of the limit of an increasing sequence of measurable sets is the limit of the volumes. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( ( F : NN
 --> dom  vol  /\  A. n  e.  NN  ( F `  n )  C_  ( F `
  ( n  +  1 ) ) ) 
 ->  ( vol `  U. ran  F )  =  sup (
 ( vol " ran  F ) ,  RR* ,  <  ) )
 
Theoremiunmbl2 22133* The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  ~<_  NN  /\  A. n  e.  A  B  e.  dom  vol )  ->  U_ n  e.  A  B  e.  dom  vol )
 
Theoremioombl1lem1 22134* Lemma for ioombl1 22138. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,) +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq 1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  H : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) ) )
 
Theoremioombl1lem2 22135* Lemma for ioombl1 22138. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,) +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq 1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  e.  RR )
 
Theoremioombl1lem3 22136* Lemma for ioombl1 22138. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,) +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq 1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ( ph  /\  n  e.  NN )  ->  ( ( ( ( abs  o.  -  )  o.  G ) `  n )  +  ( (
 ( abs  o.  -  )  o.  H ) `  n ) )  =  (
 ( ( abs  o.  -  )  o.  F ) `
  n ) )
 
Theoremioombl1lem4 22137* Lemma for ioombl1 22138. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  B  =  ( A (,) +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq 1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  ( ( vol* `  ( E  i^i  B ) )  +  ( vol* `  ( E  \  B ) ) ) 
 <_  ( ( vol* `  E )  +  C ) )
 
Theoremioombl1 22138 An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
 |-  ( A  e.  RR*  ->  ( A (,) +oo )  e.  dom  vol )
 
Theoremicombl1 22139 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A  e.  RR  ->  ( A [,) +oo )  e.  dom  vol )
 
Theoremicombl 22140 A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( A [,) B )  e.  dom  vol )
 
Theoremioombl 22141 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A (,) B )  e.  dom  vol
 
Theoremiccmbl 22142 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  e.  dom  vol )
 
Theoremiccvolcl 22143 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A [,] B ) )  e.  RR )
 
Theoremovolioo 22144 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A (,) B ) )  =  ( B  -  A ) )
 
Theoremioovolcl 22145 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A (,) B ) )  e.  RR )
 
Theoremovolfs2 22146 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  G  =  ( ( abs  o.  -  )  o.  F )   =>    |-  ( F : NN --> (  <_  i^i  ( RR  X. 
 RR ) )  ->  G  =  ( ( vol*  o.  (,) )  o.  F ) )
 
Theoremioorcl2 22147 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ( ( A (,) B )  =/=  (/)  /\  ( vol* `  ( A (,) B ) )  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR ) )
 
Theoremioorf 22148 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  F : ran  (,) --> ( 
 <_  i^i  ( RR*  X.  RR* ) )
 
Theoremioorval 22149* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( A  e.  ran  (,) 
 ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
 >. ) )
 
Theoremioorinv2 22150* The function  F is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( ( A (,) B )  =/=  (/)  ->  ( F `  ( A (,) B ) )  =  <. A ,  B >. )
 
Theoremioorinv 22151* The function  F is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( A  e.  ran  (,) 
 ->  ( (,) `  ( F `  A ) )  =  A )
 
Theoremioorcl 22152* The function  F does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( ( A  e.  ran 
 (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
 
Theoremuniiccdif 22153 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   =>    |-  ( ph  ->  ( U. ran  ( (,)  o.  F )  C_  U. ran  ( [,]  o.  F ) 
 /\  ( vol* `  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 ) )
 
Theoremuniioovol 22154* A disjoint union of open intervals has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 22130.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremuniiccvol 22155* An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 22130.) (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  ( vol* `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremuniioombllem1 22156* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   =>    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
 
Theoremuniioombllem2a 22157* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   =>    |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( ( (,) `  ( F `  z
 ) )  i^i  ( (,) `  ( G `  J ) ) )  e.  ran  (,) )
 
Theoremuniioombllem2 22158* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  H  =  ( z  e.  NN  |->  ( ( (,) `  ( F `  z ) )  i^i  ( (,) `  ( G `  J ) ) ) )   &    |-  K  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) , 
 <. 0 ,  0 >. ,  <. sup ( x ,  RR*
 ,  `'  <  ) ,  sup ( x ,  RR*
 ,  <  ) >. ) )   =>    |-  ( ( ph  /\  J  e.  NN )  ->  seq 1
 (  +  ,  ( vol*  o.  H ) )  ~~>  ( vol* `  ( ( (,) `  ( G `  J ) )  i^i  A ) ) )
 
Theoremuniioombllem3a 22159* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  ( abs `  (
 ( T `  M )  -  sup ( ran 
 T ,  RR* ,  <  ) ) )  <  C )   &    |-  K  =  U. (
 ( (,)  o.  G ) " ( 1 ...
 M ) )   =>    |-  ( ph  ->  ( K  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `  j ) ) 
 /\  ( vol* `  K )  e.  RR ) )
 
Theoremuniioombllem3 22160* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  ( abs `  (
 ( T `  M )  -  sup ( ran 
 T ,  RR* ,  <  ) ) )  <  C )   &    |-  K  =  U. (
 ( (,)  o.  G ) " ( 1 ...
 M ) )   =>    |-  ( ph  ->  ( ( vol* `  ( E  i^i  A ) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
 
Theoremuniioombllem4 22161* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  ( abs `  (
 ( T `  M )  -  sup ( ran 
 T ,  RR* ,  <  ) ) )  <  C )   &    |-  K  =  U. (
 ( (,)  o.  G ) " ( 1 ...
 M ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( abs `  ( sum_ i  e.  ( 1
 ... N ) ( vol* `  (
 ( (,) `  ( F `  i ) )  i^i  ( (,) `  ( G `  j ) ) ) )  -  ( vol* `  ( ( (,) `  ( G `  j ) )  i^i 
 A ) ) ) )  <  ( C 
 /  M ) )   &    |-  L  =  U. ( ( (,)  o.  F )
 " ( 1 ...
 N ) )   =>    |-  ( ph  ->  ( vol* `  ( K  i^i  A ) ) 
 <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
 
Theoremuniioombllem5 22162* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  ( abs `  (
 ( T `  M )  -  sup ( ran 
 T ,  RR* ,  <  ) ) )  <  C )   &    |-  K  =  U. (
 ( (,)  o.  G ) " ( 1 ...
 M ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( abs `  ( sum_ i  e.  ( 1
 ... N ) ( vol* `  (
 ( (,) `  ( F `  i ) )  i^i  ( (,) `  ( G `  j ) ) ) )  -  ( vol* `  ( ( (,) `  ( G `  j ) )  i^i 
 A ) ) ) )  <  ( C 
 /  M ) )   &    |-  L  =  U. ( ( (,)  o.  F )
 " ( 1 ...
 N ) )   =>    |-  ( ph  ->  ( ( vol* `  ( E  i^i  A ) )  +  ( vol* `  ( E  \  A ) ) ) 
 <_  ( ( vol* `  E )  +  (
 4  x.  C ) ) )
 
Theoremuniioombllem6 22163* Lemma for uniioombl 22164. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol* `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C ) )   =>    |-  ( ph  ->  (
 ( vol* `  ( E  i^i  A ) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( ( vol* `  E )  +  (
 4  x.  C ) ) )
 
Theoremuniioombl 22164* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 22129.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  U.
 ran  ( (,)  o.  F )  e.  dom  vol )
 
Theoremuniiccmbl 22165* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 22129.) (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq 1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  U.
 ran  ( [,]  o.  F )  e.  dom  vol )
 
Theoremdyadf 22166* The function  F returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  F : ( ZZ 
 X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )
 
Theoremdyadval 22167* Value of the dyadic rational function  F. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  ->  ( A F B )  =  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  +  1 )  /  ( 2 ^ B ) ) >. )
 
Theoremdyadovol 22168* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  ->  ( vol* `  ( [,] `  ( A F B ) ) )  =  ( 1 
 /  ( 2 ^ B ) ) )
 
Theoremdyadss 22169* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  ->  ( ( [,] `  ( A F C ) ) 
 C_  ( [,] `  ( B F D ) ) 
 ->  D  <_  C )
 )
 
Theoremdyaddisjlem 22170* Lemma for dyaddisj 22171. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 )
 )  /\  C  <_  D )  ->  ( ( [,] `  ( A F C ) )  C_  ( [,] `  ( B F D ) )  \/  ( [,] `  ( B F D ) ) 
 C_  ( [,] `  ( A F C ) )  \/  ( ( (,) `  ( A F C ) )  i^i  ( (,) `  ( B F D ) ) )  =  (/) ) )
 
Theoremdyaddisj 22171* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( A  e.  ran 
 F  /\  B  e.  ran 
 F )  ->  (
 ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B )  C_  ( [,] `  A )  \/  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  (/) ) )
 
Theoremdyadmaxlem 22172* Lemma for dyadmax 22173. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  -.  D  <  C )   &    |-  ( ph  ->  ( [,] `  ( A F C ) )  C_  ( [,] `  ( B F D ) ) )   =>    |-  ( ph  ->  ( A  =  B  /\  C  =  D )
 )
 
Theoremdyadmax 22173* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( ( A  C_  ran 
 F  /\  A  =/=  (/) )  ->  E. z  e.  A  A. w  e.  A  ( ( [,] `  z )  C_  ( [,] `  w )  ->  z  =  w )
 )
 
Theoremdyadmbllem 22174* Lemma for dyadmbl 22175. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   &    |-  G  =  {
 z  e.  A  |  A. w  e.  A  ( ( [,] `  z
 )  C_  ( [,] `  w )  ->  z  =  w ) }   &    |-  ( ph  ->  A  C_  ran  F )   =>    |-  ( ph  ->  U. ( [,] " A )  = 
 U. ( [,] " G ) )
 
Theoremdyadmbl 22175* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   &    |-  G  =  {
 z  e.  A  |  A. w  e.  A  ( ( [,] `  z
 )  C_  ( [,] `  w )  ->  z  =  w ) }   &    |-  ( ph  ->  A  C_  ran  F )   =>    |-  ( ph  ->  U. ( [,] " A )  e. 
 dom  vol )
 
Theoremopnmbllem 22176* Lemma for opnmbl 22177. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  ( 2 ^ y
 ) ) ,  (
 ( x  +  1 )  /  ( 2 ^ y ) )
 >. )   =>    |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremopnmbl 22177 All open sets are measurable. This proof, via dyadmbl 22175 and uniioombl 22164, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
TheoremopnmblALT 22178 All open sets are measurable. This alternative proof of opnmbl 22177 is significantly shorter, at the expense of invoking countable choice ax-cc 8806. (This was also the original proof before the current opnmbl 22177 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremsubopnmbl 22179 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  J  =  ( (
 topGen `  ran  (,) )t  A )   =>    |-  ( ( A  e.  dom 
 vol  /\  B  e.  J )  ->  B  e.  dom  vol )
 
Theoremvolsup2 22180* The volume of  A is the supremum of the sequence  vol* `  ( A  i^i  ( -u n [,] n ) ) of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR  /\  B  <  ( vol `  A ) )  ->  E. n  e.  NN  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
 
Theoremvolcn 22181* The function formed by restricting a measurable set to a closed interval with a varying endpoint produces an increasing continuous function on the reals. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  F  =  ( x  e.  RR  |->  ( vol `  ( A  i^i  ( B [,] x ) ) ) )   =>    |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR )  ->  F  e.  ( RR -cn-> RR ) )
 
Theoremvolivth 22182* The Intermediate Value Theorem for the Lebesgue volume function. For any positive  B  <_  ( vol `  A ), there is a measurable subset of  A whose volume is  B. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  (
 0 [,] ( vol `  A ) ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x )  =  B )
 )
 
Theoremvitalilem1 22183* Lemma for vitali 22188. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  (
 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y
 )  e.  QQ ) }   =>    |- 
 .~  Er  ( 0 [,] 1 )
 
Theoremvitalilem2 22184* Lemma for vitali 22188. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  (
 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y
 )  e.  QQ ) }   &    |-  S  =  ( ( 0 [,] 1 )
 /.  .~  )   &    |-  ( ph  ->  F  Fn  S )   &    |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z
 ) )   &    |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )   &    |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F }
 )   &    |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \ 
 dom  vol ) )   =>    |-  ( ph  ->  ( ran  F  C_  (
 0 [,] 1 )  /\  ( 0 [,] 1
 )  C_  U_ m  e. 
 NN  ( T `  m )  /\  U_ m  e.  NN  ( T `  m )  C_  ( -u 1 [,] 2 ) ) )
 
Theoremvitalilem3 22185* Lemma for vitali 22188. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  (
 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y
 )  e.  QQ ) }   &    |-  S  =  ( ( 0 [,] 1 )
 /.  .~  )   &    |-  ( ph  ->  F  Fn  S )   &    |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z
 ) )   &    |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )   &    |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F }
 )   &    |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \ 
 dom  vol ) )   =>    |-  ( ph  -> Disj  m  e.  NN  ( T `  m ) )
 
Theoremvitalilem4 22186* Lemma for vitali 22188. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  (
 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y
 )  e.  QQ ) }   &    |-  S  =  ( ( 0 [,] 1 )
 /.  .~  )   &    |-  ( ph  ->  F  Fn  S )   &    |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z
 ) )   &    |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )   &    |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F }
 )   &    |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \ 
 dom  vol ) )   =>    |-  ( ( ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  0 )
 
Theoremvitalilem5 22187* Lemma for vitali 22188. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  (
 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y
 )  e.  QQ ) }   &    |-  S  =  ( ( 0 [,] 1 )
 /.  .~  )   &    |-  ( ph  ->  F  Fn  S )   &    |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z
 ) )   &    |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )   &    |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F }
 )   &    |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \ 
 dom  vol ) )   =>    |-  -.  ph
 
Theoremvitali 22188 If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )
 
13.2.2  Lebesgue integration
 
13.2.2.1  Lesbesgue integral
 
Syntaxcmbf 22189 Extend class notation with the class of measurable functions.
 class MblFn
 
Syntaxcitg1 22190 Extend class notation with the Lebesgue integral for simple functions.
 class  S.1
 
Syntaxcitg2 22191 Extend class notation with the Lebesgue integral for nonnegative functions.
 class  S.2
 
Syntaxcibl 22192 Extend class notation with the class of integrable functions.
 class  L^1
 
Syntaxcitg 22193 Extend class notation with the general Lebesgue integral.
 class  S. A B  _d x
 
Definitiondf-mbf 22194* Define the class of measurable functions on the reals. A real function is measurable if the preimage of every open interval is a measurable set (see ismbl 22103) and a complex function is measurable if the real and imaginary parts of the function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |- MblFn  =  { f  e.  ( CC  ^pm  RR )  | 
 A. x  e.  ran  (,) ( ( `' ( Re  o.  f ) " x )  e.  dom  vol  /\  ( `' ( Im 
 o.  f ) " x )  e.  dom  vol ) }
 
Definitiondf-itg1 22195* Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |- 
 S.1  =  ( f  e.  { g  e. MblFn  |  ( g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } )
 ) )  e.  RR ) }  |->  sum_ x  e.  ( ran  f  \  { 0 } )
 ( x  x.  ( vol `  ( `' f " { x } )
 ) ) )
 
Definitiondf-itg2 22196* Define the Lebesgue integral for nonnegative functions. A nonnegative function's integral is the supremum of the integrals of all simple functions that are less than the input function. Note that this may be +oo for functions that take the value +oo on a set of positive measure or functions that are bounded below by a positive number on a set of infinite measure. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |- 
 S.2  =  ( f  e.  ( ( 0 [,] +oo )  ^m  RR )  |-> 
 sup ( { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
 ) ) } ,  RR*
 ,  <  ) )
 
Definitiondf-ibl 22197* Define the class of integrable functions on the reals. A function is integrable if it is measurable and the integrals of the pieces of the function are all finite. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L^1  =  {
 f  e. MblFn  |  A. k  e.  ( 0 ... 3
 ) ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( ( f `
  x )  /  ( _i ^ k ) ) )  /  y ]_ if ( ( x  e.  dom  f  /\  0  <_  y ) ,  y ,  0 ) ) )  e.  RR }
 
Definitiondf-itg 22198* Define the full Lebesgue integral, for complex-valued functions to  RR. The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of  x ^ 2 from  0 to  1 is  S. ( 0 [,] 1 ) ( x ^ 2 )  _d x  =  ( 1  /  3 ). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 22196 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 22196 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
 |- 
 S. A B  _d x  =  sum_ k  e.  ( 0 ... 3
 ) ( ( _i
 ^ k )  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^ k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 ) ) ) )
 
Theoremismbf1 22199* The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 22203 and ismbfcn 22204 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  <->  ( F  e.  ( CC  ^pm  RR )  /\  A. x  e.  ran  (,) ( ( `' ( Re  o.  F ) " x )  e.  dom  vol  /\  ( `' ( Im 
 o.  F ) " x )  e.  dom  vol ) ) )
 
Theoremmbff 22200 A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  F : dom  F --> CC )
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