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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvoliun 19401 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 19402* Lemma for volsup 19403. (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 19403* 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 19404* 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 19405* Lemma for ioombl1 19409. (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 19406* Lemma for ioombl1 19409. (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 19407* Lemma for ioombl1 19409. (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 19408* Lemma for ioombl1 19409. (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 19409 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 19410 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
 
Theoremicombl 19411 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 19412 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A (,) B )  e.  dom  vol
 
Theoremiccmbl 19413 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 19414 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 19415 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 19416 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 19417 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 19418 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 19419* 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 19420* 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 19421* 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 19422* 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 19423 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 19424* 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 19401.) 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 19425* 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 19401.) (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 19426* Lemma for uniioombl 19434. (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 19427* Lemma for uniioombl 19434. (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 19428* Lemma for uniioombl 19434. (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 19429* Lemma for uniioombl 19434. (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 19430* Lemma for uniioombl 19434. (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 19431* Lemma for uniioombl 19434. (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 19432* Lemma for uniioombl 19434. (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 19433* Lemma for uniioombl 19434. (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 19434* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 19400.) 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 19435* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 19400.) (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 19436* 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 19437* 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 19438* 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 19439* 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 19440* Lemma for dyaddisj 19441. (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 19441* 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 19442* Lemma for dyadmax 19443. (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 19443* 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 19444* Lemma for dyadmbl 19445. (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 19445* 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 19446* Lemma for opnmbl 19447. (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 19447 All open sets are measurable. This proof, via dyadmbl 19445 and uniioombl 19434, 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 19448 All open sets are measurable. This alternative proof of opnmbl 19447 is significantly shorter, at the expense of invoking countable choice ax-cc 8271. (This was also the original proof before the current opnmbl 19447 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremsubopnmbl 19449 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 19450* 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 19451* 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 19452* 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 19453* Lemma for vitali 19458. (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 19454* Lemma for vitali 19458. (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 19455* Lemma for vitali 19458. (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 19456* Lemma for vitali 19458. (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 19457* Lemma for vitali 19458. (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 19458 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 19459 Extend class notation with the class of measurable functions.
 class MblFn
 
Syntaxcitg1 19460 Extend class notation with the Lebesgue integral for simple functions.
 class  S.1
 
Syntaxcitg2 19461 Extend class notation with the Lebesgue integral for nonnegative functions.
 class  S.2
 
Syntaxcibl 19462 Extend class notation with the class of integrable functions.
 class  L ^1
 
Syntaxcitg 19463 Extend class notation with the general Lebesgue integral.
 class  S. A B  _d x
 
Syntaxcdit 19464 Extend class notation with the directed integral.
 class  S__ [ A  ->  B ] C  _d x
 
Definitiondf-mbf 19465* 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 19375) 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 19466* 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 19467* 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 19468* 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 19469* 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 19467 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 19467 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 19470 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 19471* The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19475 and ismbfcn 19476 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 19472 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  F : dom  F --> CC )
 
Theoremmbfdm 19473 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  dom 
 F  e.  dom  vol )
 
Theoremmbfconstlem 19474 Lemma for mbfconst 19480. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  C  e.  RR )  ->  ( `' ( A  X.  { C }
 ) " B )  e. 
 dom  vol )
 
Theoremismbf 19475* 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 19375. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol ) )
 
Theoremismbfcn 19476 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 19477 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 19478 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 19479 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 19480 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 19481 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( A  e.  dom  vol 
 ->  (  _I  |`  A )  e. MblFn )
 
Theoremmbfmptcl 19482* 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 19483* 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 19484* 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 19485* 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 19499. (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 19486* 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 19487* Lemma for mbfeqa 19488. (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 19488* 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 19489 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 19490 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 )
 
Theoremmbfss 19491* Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  dom  vol )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  e.  V )   &    |-  ( ( ph  /\  x  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  B  |->  C )  e. MblFn )
 
Theoremmbfmulc2lem 19492 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : A --> RR )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F  x.  F )  e. MblFn )
 
Theoremmbfmulc2re 19493 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : A --> CC )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F  x.  F )  e. MblFn )
 
Theoremmbfmax 19494* The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G : A --> RR )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  H  =  ( x  e.  A  |->  if ( ( F `  x )  <_  ( G `
  x ) ,  ( G `  x ) ,  ( F `  x ) ) )   =>    |-  ( ph  ->  H  e. MblFn )
 
Theoremmbfneg 19495* The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  -u B )  e. MblFn )
 
Theoremmbfpos 19496* The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  if (
 0  <_  B ,  B ,  0 )
 )  e. MblFn )
 
Theoremmbfposr 19497* Converse to mbfpos 19496. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  if ( 0  <_  B ,  B ,  0 ) )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  if ( 0  <_  -u B ,  -u B ,  0 ) )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn
 )
 
Theoremmbfposb 19498* A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e. MblFn  <->  ( ( x  e.  A  |->  if (
 0  <_  B ,  B ,  0 )
 )  e. MblFn  /\  ( x  e.  A  |->  if (
 0  <_  -u B ,  -u B ,  0 ) )  e. MblFn ) )
 )
 
Theoremismbf3d 19499* Simplified form of ismbfd 19485. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F : A --> RR )   &    |-  (
 ( ph  /\  x  e. 
 RR )  ->  ( `' F " ( x (,)  +oo ) )  e. 
 dom  vol )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremmbfimaopnlem 19500* Lemma for mbfimaopn 19501. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  G  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   &    |-  B  =  ( (,) " ( QQ 
 X.  QQ ) )   &    |-  K  =  ran  ( x  e.  B ,  y  e.  B  |->  ( x  X.  y ) )   =>    |-  ( ( F  e. MblFn  /\  A  e.  J )  ->  ( `' F " A )  e.  dom  vol )
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