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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfiniunmbl 18901* 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 18902 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 18903 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 18904* 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 18905* 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 18906* 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 18907* Lemma for voliun 18911. (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 18908* Lemma for voliun 18911. (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 18909* Lemma for voliun 18911. (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 18910 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 18911 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 18912* Lemma for volsup 18913. (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 18913* 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 18914* 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 18915* Lemma for ioombl1 18919. (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 18916* Lemma for ioombl1 18919. (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 18917* Lemma for ioombl1 18919. (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 18918* Lemma for ioombl1 18919. (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 18919 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 18920 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
 
Theoremicombl 18921 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 18922 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A (,) B )  e.  dom  vol
 
Theoremiccmbl 18923 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 18924 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 18925 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 ) )
 
Theoremovolfs2 18926 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 18927 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 18928 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 18929* 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 18930* 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 18931* 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 18932* 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 18933 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 18934* An 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 18911.) 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 18935* 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 18911.) (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 18936* Lemma for uniioombl 18944. (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 18937* Lemma for uniioombl 18944. (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 18938* Lemma for uniioombl 18944. (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 18939* Lemma for uniioombl 18944. (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 18940* Lemma for uniioombl 18944. (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 18941* Lemma for uniioombl 18944. (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 18942* Lemma for uniioombl 18944. (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 18943* Lemma for uniioombl 18944. (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 18944* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18910.) 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 18945* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18910.) (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 18946* 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 18947* 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 18948* 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 18949* 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 18950* Lemma for dyaddisj 18951. (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 18951* 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 18952* Lemma for dyadmax 18953. (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 18953* 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 18954* Lemma for dyadmbl 18955. (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 18955* 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 18956* Lemma for opnmbl 18957. (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 18957 All open sets are measurable. This proof, via dyadmbl 18955 and uniioombl 18944, 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 18958 All open sets are measurable. This alternative proof of opnmbl 18957 is significantly shorter, at the expense of invoking countable choice ax-cc 8061. (This was also the original proof before the current opnmbl 18957 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremsubopnmbl 18959 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 18960* 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 18961* 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 18962* 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 18963* Lemma for vitali 18968. (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 18964* Lemma for vitali 18968. (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 18965* Lemma for vitali 18968. (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 18966* Lemma for vitali 18968. (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 18967* Lemma for vitali 18968. (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 18968 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 )
 
12.2.2  Lebesgue integration
 
Syntaxcmbf 18969 Extend class notation with the class of measurable functions.
 class MblFn
 
Syntaxcitg1 18970 Extend class notation with the Lebesgue integral for simple functions.
 class  S.1
 
Syntaxcitg2 18971 Extend class notation with the Lebesgue integral for nonnegative functions.
 class  S.2
 
Syntaxcibl 18972 Extend class notation with the class of integrable functions.
 class  L ^1
 
Syntaxcitg 18973 Extend class notation with the general Lebesgue integral.
 class  S. A B  _d x
 
Syntaxcdit 18974 Extend class notation with the directed integral.
 class  S__ [ A  ->  B ] C  _d x
 
Definitiondf-mbf 18975* 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 18885) 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 18976* 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 18977* 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  o R  <_  f  /\  x  =  ( S.1 `  g
 ) ) } ,  RR*
 ,  <  ) )
 
Definitiondf-ibl 18978* 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 18979* 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 18977 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 18977 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 ) ) ) )
 
Definitiondf-ditg 18980 Define the directed integral, which is just a regular integral but with a sign change when the limits are interchanged. The  A and  B here are the lower and upper limits of the integral, usually written as a subscript and superscript next to the integral sign. We define the region of integration to be an open interval instead of closed so that we can use  +oo ,  -oo for limits and also integrate up to a singularity at an endpoint. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |- 
 S__ [ A  ->  B ] C  _d x  =  if ( A  <_  B ,  S. ( A (,) B ) C  _d x ,  -u S. ( B (,) A ) C  _d x )
 
Theoremismbf1 18981* The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 18985 and ismbfcn 18986 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 18982 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  F : dom  F --> CC )
 
Theoremmbfdm 18983 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  dom 
 F  e.  dom  vol )
 
Theoremmbfconstlem 18984 Lemma for mbfconst 18990. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  C  e.  RR )  ->  ( `' ( A  X.  { C }
 ) " B )  e. 
 dom  vol )
 
Theoremismbf 18985* The predicate " F is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 18885. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol ) )
 
Theoremismbfcn 18986 A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F : A --> CC  ->  ( F  e. MblFn  <->  (
 ( Re  o.  F )  e. MblFn  /\  ( Im 
 o.  F )  e. MblFn
 ) ) )
 
Theoremmbfima 18987 Definitional property of a measurable function: the preimage of an open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( F  e. MblFn  /\  F : A --> RR )  ->  ( `' F "
 ( B (,) C ) )  e.  dom  vol )
 
Theoremmbfimaicc 18988 The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( B [,] C ) )  e.  dom  vol )
 
Theoremmbfimasn 18989 The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ( F  e. MblFn  /\  F : A --> RR  /\  B  e.  RR )  ->  ( `' F " { B } )  e. 
 dom  vol )
 
Theoremmbfconst 18990 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  CC )  ->  ( A  X.  { B } )  e. MblFn
 )
 
Theoremmbfid 18991 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( A  e.  dom  vol 
 ->  (  _I  |`  A )  e. MblFn )
 
Theoremmbfmptcl 18992* Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   =>    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )
 
Theoremmbfdm2 18993* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   =>    |-  ( ph  ->  A  e.  dom 
 vol )
 
Theoremismbfcn2 18994* A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e. MblFn  <->  ( ( x  e.  A  |->  ( Re
 `  B ) )  e. MblFn  /\  ( x  e.  A  |->  ( Im `  B ) )  e. MblFn
 ) ) )
 
Theoremismbfd 18995* Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 19009. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F : A --> RR )   &    |-  (
 ( ph  /\  x  e.  RR* )  ->  ( `' F " ( x (,)  +oo ) )  e. 
 dom  vol )   &    |-  ( ( ph  /\  x  e.  RR* )  ->  ( `' F "
 (  -oo (,) x ) )  e.  dom  vol )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremismbf2d 18996* Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ( ph  /\  x  e.  RR )  ->  ( `' F "
 ( x (,)  +oo ) )  e.  dom  vol )   &    |-  ( ( ph  /\  x  e.  RR )  ->  ( `' F "
 (  -oo (,) x ) )  e.  dom  vol )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremmbfeqalem 18997* Lemma for mbfeqa 18998. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( B  \  A ) )  ->  C  =  D )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  B )  ->  D  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  B  |->  C )  e. MblFn  <->  ( x  e.  B  |->  D )  e. MblFn
 ) )
 
Theoremmbfeqa 18998* If two functions are equal almost everywhere, then one is measurable iff the other is. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( B  \  A ) )  ->  C  =  D )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  B )  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  B  |->  C )  e. MblFn  <->  ( x  e.  B  |->  D )  e. MblFn
 ) )
 
Theoremmbfres 18999 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ( F  e. MblFn  /\  A  e.  dom  vol )  ->  ( F  |`  A )  e. MblFn )
 
Theoremmbfres2 19000 Measurability of a piecewise function: if  F is measurable on subsets  B and  C of its domain, and these pieces make up all of  A, then  F is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  ( F  |`  B )  e. MblFn )   &    |-  ( ph  ->  ( F  |`  C )  e. MblFn )   &    |-  ( ph  ->  ( B  u.  C )  =  A )   =>    |-  ( ph  ->  F  e. MblFn )
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