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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovolshftlem2 18701* Lemma for ovolshft 18702. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   &    |-  M  =  { y  e.  RR*  | 
 E. f  e.  (
 (  <_  i^i  ( RR 
 X.  RR ) )  ^m  NN ) ( B  C_  U.
 ran  ( (,)  o.  f )  /\  y  = 
 sup ( ran  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) }   =>    |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( A  C_  U. ran  ( (,)  o.  g ) 
 /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
 RR* ,  <  ) ) }  C_  M )
 
Theoremovolshft 18702* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   =>    |-  ( ph  ->  ( vol * `  A )  =  ( vol * `  B ) )
 
Theoremsca2rab 18703* If  B is a scale of  A by  C, then  A is a scale of  B by  1  /  C. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   =>    |-  ( ph  ->  A  =  { y  e. 
 RR  |  ( ( 1  /  C )  x.  y )  e.  B } )
 
Theoremovolscalem1 18704* Lemma for ovolsca 18706. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  G  =  ( n  e.  NN  |->  <.
 ( ( 1st `  ( F `  n ) ) 
 /  C ) ,  ( ( 2nd `  ( F `  n ) ) 
 /  C ) >. )   &    |-  ( ph  ->  F : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  A 
 C_  U. ran  ( (,) 
 o.  F ) )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  A )  +  ( C  x.  R ) ) )   =>    |-  ( ph  ->  ( vol * `  B ) 
 <_  ( ( ( vol
 * `  A )  /  C )  +  R ) )
 
Theoremovolscalem2 18705* Lemma for ovolshft 18702. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  B ) 
 <_  ( ( vol * `  A )  /  C ) )
 
Theoremovolsca 18706* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  B )  =  ( ( vol
 * `  A )  /  C ) )
 
Theoremovolicc1 18707* The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  G  =  ( n  e.  NN  |->  if ( n  =  1 ,  <. A ,  B >. , 
 <. 0 ,  0 >.
 ) )   =>    |-  ( ph  ->  ( vol * `  ( A [,] B ) ) 
 <_  ( B  -  A ) )
 
Theoremovolicc2lem1 18708* Lemma for ovolicc2 18713. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   =>    |-  ( ( ph  /\  X  e.  U ) 
 ->  ( P  e.  X  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X ) ) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
 
Theoremovolicc2lem2 18709* Lemma for ovolicc2 18713. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   =>    |-  (
 ( ph  /\  ( N  e.  NN  /\  -.  N  e.  W )
 )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  <_  B )
 
Theoremovolicc2lem3 18710* Lemma for ovolicc2 18713. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   =>    |-  (
 ( ph  /\  ( N  e.  { n  e. 
 NN  |  A. m  e.  W  n  <_  m }  /\  P  e.  { n  e.  NN  |  A. m  e.  W  n  <_  m } ) ) 
 ->  ( N  =  P  <->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  =  ( 2nd `  ( F `  ( G `  ( K `  P ) ) ) ) ) )
 
Theoremovolicc2lem4 18711* Lemma for ovolicc2 18713. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   &    |-  M  =  sup ( W ,  RR ,  `'  <  )   =>    |-  ( ph  ->  ( B  -  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremovolicc2lem5 18712* Lemma for ovolicc2 18713. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   =>    |-  ( ph  ->  ( B  -  A )  <_  sup ( ran  S ,  RR*
 ,  <  ) )
 
Theoremovolicc2 18713* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( ( A [,] B )  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran 
 seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ,  RR* ,  <  ) ) }   =>    |-  ( ph  ->  ( B  -  A )  <_  ( vol * `  ( A [,] B ) ) )
 
Theoremovolicc 18714 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol * `  ( A [,] B ) )  =  ( B  -  A ) )
 
Theoremovolicopnf 18715 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo )
 )  =  +oo )
 
Theoremovolre 18716 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( vol * `  RR )  =  +oo
 
Theoremismbl 18717* The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set  x in a "nice" way, that is, if the measure of the pieces  x  i^i  A and  x  \  A sum up to the measure of 
x (assuming that the measure of 
x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol * `  ( x  \  A ) ) ) ) ) )
 
Theoremismbl2 18718* From ovolun 18690, it suffices to show that the measure of  x is at least the sum of the measures of  x  i^i  A and  x  \  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol * `  x )  e.  RR  ->  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
 * `  ( x  \  A ) ) ) 
 <_  ( vol * `  x ) ) ) )
 
Theoremvolres 18719 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol  =  ( vol *  |`  dom  vol )
 
Theoremvolf 18720 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol : dom  vol --> ( 0 [,]  +oo )
 
Theoremmblvol 18721 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( vol `  A )  =  ( vol * `
  A ) )
 
Theoremmblss 18722 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  A  C_  RR )
 
Theoremmblsplit 18723 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  C_  RR  /\  ( vol * `  B )  e.  RR )  ->  ( vol * `  B )  =  ( ( vol * `  ( B  i^i  A ) )  +  ( vol
 * `  ( B  \  A ) ) ) )
 
Theoremcmmbl 18724 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( RR  \  A )  e.  dom  vol )
 
Theoremnulmbl 18725 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  ->  A  e.  dom 
 vol )
 
Theoremnulmbl2 18726* A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has  vol * ( A )  =  0 while "outer measure zero" means that for any  x there is a  y containing  A with volume less than  x. Assuming AC, these notions are equivalent (because the intersection of all such  y is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A. x  e.  RR+  E. y  e.  dom  vol ( A  C_  y  /\  ( vol * `  y )  <_  x ) 
 ->  A  e.  dom  vol )
 
Theoremunmbl 18727 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  u.  B )  e.  dom  vol )
 
Theoremshftmbl 18728* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR )  ->  { x  e. 
 RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
 
Theorem0mbl 18729 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  (/)  e.  dom  vol
 
Theoremrembl 18730 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |- 
 RR  e.  dom  vol
 
Theoreminmbl 18731 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  i^i  B )  e.  dom  vol )
 
Theoremdifmbl 18732 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  \  B )  e.  dom  vol )
 
Theoremfiniunmbl 18733* 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 18734 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 18735 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 18736* 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 18737* 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 18738* 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 18739* Lemma for voliun 18743. (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 18740* Lemma for voliun 18743. (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 18741* Lemma for voliun 18743. (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 18742 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 18743 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 18744* Lemma for volsup 18745. (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 18745* 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 18746* 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 18747* Lemma for ioombl1 18751. (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 18748* Lemma for ioombl1 18751. (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 18749* Lemma for ioombl1 18751. (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 18750* Lemma for ioombl1 18751. (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 18751 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 18752 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
 
Theoremicombl 18753 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 18754 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A (,) B )  e.  dom  vol
 
Theoremiccmbl 18755 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 18756 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 18757 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 18758 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 18759 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 18760 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 18761* 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 18762* 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 18763* 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 18764* 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 18765 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 18766* 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 18743.) 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 18767* 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 18743.) (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 18768* Lemma for uniioombl 18776. (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 18769* Lemma for uniioombl 18776. (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 18770* Lemma for uniioombl 18776. (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 18771* Lemma for uniioombl 18776. (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 18772* Lemma for uniioombl 18776. (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 18773* Lemma for uniioombl 18776. (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 18774* Lemma for uniioombl 18776. (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 18775* Lemma for uniioombl 18776. (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 18776* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18742.) 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 18777* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18742.) (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 18778* 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 18779* 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 18780* 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 18781* 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 18782* Lemma for dyaddisj 18783. (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 18783* 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 18784* Lemma for dyadmax 18785. (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 18785* 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 18786* Lemma for dyadmbl 18787. (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 18787* 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 18788* Lemma for opnmbl 18789. (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 18789 All open sets are measurable. This proof, via dyadmbl 18787 and uniioombl 18776, 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 18790 All open sets are measurable. This alternative proof of opnmbl 18789 is significantly shorter, at the expense of invoking countable choice ax-cc 7945. (This was also the original proof before the current opnmbl 18789 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)
 |-  ( A  e.  ( topGen `
  ran  (,) )  ->  A  e.  dom  vol )
 
Theoremsubopnmbl 18791 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 18792* 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 18793* 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 18794* 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 18795* Lemma for vitali 18800. (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 18796* Lemma for vitali 18800. (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 18797* Lemma for vitali 18800. (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 18798* Lemma for vitali 18800. (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 18799* Lemma for vitali 18800. (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 18800 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 )
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