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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuniioombllem3a 18901* Lemma for uniioombl 18906. (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 18902* Lemma for uniioombl 18906. (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 18903* Lemma for uniioombl 18906. (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 18904* Lemma for uniioombl 18906. (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 18905* Lemma for uniioombl 18906. (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 18906* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18872.) 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 18907* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18872.) (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 18908* 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 18909* 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 18910* 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 18911* 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 18912* Lemma for dyaddisj 18913. (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 18913* 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 18914* Lemma for dyadmax 18915. (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 18915* 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 18916* Lemma for dyadmbl 18917. (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 18917* 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 18918* Lemma for opnmbl 18919. (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 18919 All open sets are measurable. This proof, via dyadmbl 18917 and uniioombl 18906, 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 18920 All open sets are measurable. This alternative proof of opnmbl 18919 is significantly shorter, at the expense of invoking countable choice ax-cc 8029. (This was also the original proof before the current opnmbl 18919 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremsubopnmbl 18921 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 18922* 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 18923* 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 18924* 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 18925* Lemma for vitali 18930. (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 18926* Lemma for vitali 18930. (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 18927* Lemma for vitali 18930. (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 18928* Lemma for vitali 18930. (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 18929* Lemma for vitali 18930. (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 18930 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 18931 Extend class notation with the class of measurable functions.
 class MblFn
 
Syntaxcitg1 18932 Extend class notation with the Lebesgue integral for simple functions.
 class  S.1
 
Syntaxcitg2 18933 Extend class notation with the Lebesgue integral for nonnegative functions.
 class  S.2
 
Syntaxcibl 18934 Extend class notation with the class of integrable functions.
 class  L ^1
 
Syntaxcitg 18935 Extend class notation with the general Lebesgue integral.
 class  S. A B  _d x
 
Syntaxcdit 18936 Extend class notation with the directed integral.
 class  S__ [ A  ->  B ] C  _d x
 
Definitiondf-mbf 18937* 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 18847) 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 18938* 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 18939* 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 18940* 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 18941* 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 18939 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 18939 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 18942 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 18943* The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 18947 and ismbfcn 18948 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 18944 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  F : dom  F --> CC )
 
Theoremmbfdm 18945 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  dom 
 F  e.  dom  vol )
 
Theoremmbfconstlem 18946 Lemma for mbfconst 18952. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  C  e.  RR )  ->  ( `' ( A  X.  { C }
 ) " B )  e. 
 dom  vol )
 
Theoremismbf 18947* 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 18847. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol ) )
 
Theoremismbfcn 18948 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 18949 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 18950 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 18951 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 18952 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 18953 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( A  e.  dom  vol 
 ->  (  _I  |`  A )  e. MblFn )
 
Theoremmbfmptcl 18954* 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 18955* 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 18956* 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 18957* 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 18971. (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 18958* 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 18959* Lemma for mbfeqa 18960. (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 18960* 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 18961 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 18962 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 18963* 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 18964 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 18965 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 18966* 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 18967* 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 18968* 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 18969* Converse to mbfpos 18968. (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 18970* 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 18971* Simplified form of ismbfd 18957. (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 18972* Lemma for mbfimaopn 18973. (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 )
 
Theoremmbfimaopn 18973 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 18975, which explains why  A  e.  dom  vol is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( F  e. MblFn  /\  A  e.  J )  ->  ( `' F " A )  e.  dom  vol )
 
Theoremmbfimaopn2 18974 The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  B )   =>    |-  ( ( ( F  e. MblFn  /\  F : A --> B  /\  B  C_  CC )  /\  C  e.  K )  ->  ( `' F " C )  e.  dom  vol )
 
Theoremcncombf 18975 The composition of a continuous function with a measurable function is measurable. (More generally,  G can be a Borel-measurable function, but notably the condition that  G be only measurable is too weak, the usual counterexample taking 
G to be the Cantor function and  F the indicator function of the  G-image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( F  e. MblFn  /\  F : A --> B  /\  G  e.  ( B -cn->
 CC ) )  ->  ( G  o.  F )  e. MblFn )
 
Theoremcnmbf 18976 A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
 |-  ( ( A  e.  dom 
 vol  /\  F  e.  ( A -cn-> CC ) )  ->  F  e. MblFn )
 
Theoremmbfaddlem 18977 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   =>    |-  ( ph  ->  ( F  o F  +  G )  e. MblFn )
 
Theoremmbfadd 18978 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   =>    |-  ( ph  ->  ( F  o F  +  G )  e. MblFn )
 
Theoremmbfsub 18979 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   =>    |-  ( ph  ->  ( F  o F  -  G )  e. MblFn )
 
Theoremmbfmulc2 18980* A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )
 
Theoremmbfsup 18981* The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems,  B
( n ,  x
) is a function of both  n and  x, since it is an  n-indexed sequence of functions on  x. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  ( x  e.  A  |->  sup ( ran  (  n  e.  Z  |->  B ) ,  RR ,  <  ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ( ph  /\  ( n  e.  Z  /\  x  e.  A ) )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  RR  A. n  e.  Z  B  <_  y )   =>    |-  ( ph  ->  G  e. MblFn )
 
Theoremmbfinf 18982* The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  ( x  e.  A  |->  sup ( ran  (  n  e.  Z  |->  B ) ,  RR ,  `'  <  ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ( ph  /\  ( n  e.  Z  /\  x  e.  A ) )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  RR  A. n  e.  Z  y 
 <_  B )   =>    |-  ( ph  ->  G  e. MblFn )
 
Theoremmbflimsup 18983* The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 9-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  ( x  e.  A  |->  (
 limsup `  ( n  e.  Z  |->  B ) ) )   &    |-  H  =  ( m  e.  RR  |->  sup ( ( ( ( n  e.  Z  |->  B ) " ( m [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( limsup `
  ( n  e.  Z  |->  B ) )  e.  RR )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ( ph  /\  ( n  e.  Z  /\  x  e.  A ) )  ->  B  e.  RR )   =>    |-  ( ph  ->  G  e. MblFn )
 
Theoremmbflimlem 18984* The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( n  e.  Z  |->  B )  ~~>  C )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  (
 ( ph  /\  ( n  e.  Z  /\  x  e.  A ) )  ->  B  e.  RR )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn
 )
 
Theoremmbflim 18985* The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( n  e.  Z  |->  B )  ~~>  C )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  (
 ( ph  /\  ( n  e.  Z  /\  x  e.  A ) )  ->  B  e.  V )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn
 )
 
Syntaxc0p 18986 Extend class notation to include the zero polynomial.
 class 
 0 p
 
Definitiondf-0p 18987 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  0 p  =  ( CC  X.  { 0 } )
 
Theorem0pval 18988 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  CC  ->  ( 0 p `  A )  =  0
 )
 
Theorem0plef 18989 Two ways to say that the function 
F on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( F : RR --> ( 0 [,)  +oo ) 
 <->  ( F : RR --> RR  /\  0 p  o R  <_  F ) )
 
Theorem0pledm 18990 Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  F  Fn  A )   =>    |-  ( ph  ->  ( 0 p  o R  <_  F  <->  ( A  X.  { 0 } )  o R  <_  F )
 )
 
Theoremisi1f 18991 The predicate " F is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom  F  e.  dom  S.1 to represent this concept because  S.1 is the first preparation function for our final definition  S. (see df-itg 18941); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR --> RR  /\  ran 
 F  e.  Fin  /\  ( vol `  ( `' F " ( RR  \  { 0 } )
 ) )  e.  RR ) ) )
 
Theoremi1fmbf 18992 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F  e. MblFn )
 
Theoremi1ff 18993 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F : RR --> RR )
 
Theoremi1frn 18994 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ran  F  e.  Fin )
 
Theoremi1fima 18995 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ( `' F " A )  e.  dom  vol )
 
Theoremi1fima2 18996 Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  -.  0  e.  A )  ->  ( vol `  ( `' F " A ) )  e. 
 RR )
 
Theoremi1fima2sn 18997 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  A  e.  ( B  \  { 0 } ) )  ->  ( vol `  ( `' F " { A } )
 )  e.  RR )
 
Theoremi1fd 18998* A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  ran  F  e.  Fin )   &    |-  ( ( ph  /\  x  e.  ( ran 
 F  \  { 0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )   &    |-  ( ( ph  /\  x  e.  ( ran 
 F  \  { 0 } ) )  ->  ( vol `  ( `' F " { x }
 ) )  e.  RR )   =>    |-  ( ph  ->  F  e.  dom  S.1 )
 
Theoremi1f0rn 18999 Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  0  e.  ran  F )
 
Theoremitg1val 19000* The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  { 0 } )
 ( x  x.  ( vol `  ( `' F " { x } )
 ) ) )
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