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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmbfss 19001* 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 19002 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 19003 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 19004* 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 19005* 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 19006* 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 19007* Converse to mbfpos 19006. (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 19008* 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 19009* Simplified form of ismbfd 18995. (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 19010* Lemma for mbfimaopn 19011. (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 19011 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 19013, 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 19012 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 19013 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 19014 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 19015 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 19016 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 19017 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 19018* 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 19019* 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 19020* 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 19021* 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 19022* 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 19023* 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 19024 Extend class notation to include the zero polynomial.
 class 
 0 p
 
Definitiondf-0p 19025 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  0 p  =  ( CC  X.  { 0 } )
 
Theorem0pval 19026 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  CC  ->  ( 0 p `  A )  =  0
 )
 
Theorem0plef 19027 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 19028 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 19029 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 18979); 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 19030 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F  e. MblFn )
 
Theoremi1ff 19031 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 19032 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ran  F  e.  Fin )
 
Theoremi1fima 19033 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 19034 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 19035 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 19036* 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 19037 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 19038* 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 } )
 ) ) )
 
Theoremitg1val2 19039* The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) ) 
 ->  ( S.1 `  F )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x }
 ) ) ) )
 
Theoremitg1cl 19040 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ( S.1 `  F )  e.  RR )
 
Theoremitg1ge0 19041 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  0 p  o R  <_  F )  -> 
 0  <_  ( S.1 `  F ) )
 
Theoremi1f0 19042 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( RR  X.  {
 0 } )  e. 
 dom  S.1
 
Theoremitg10 19043 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( S.1 `  ( RR  X.  { 0 } ) )  =  0
 
Theoremi1f1lem 19044* Lemma for i1f1 19045 and itg11 19046. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  F  =  ( x  e.  RR  |->  if ( x  e.  A , 
 1 ,  0 ) )   =>    |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol  ->  ( `' F " { 1 } )  =  A ) )
 
Theoremi1f1 19045* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  F  =  ( x  e.  RR  |->  if ( x  e.  A , 
 1 ,  0 ) )   =>    |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR )  ->  F  e.  dom  S.1 )
 
Theoremitg11 19046* The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  F  =  ( x  e.  RR  |->  if ( x  e.  A , 
 1 ,  0 ) )   =>    |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR )  ->  ( S.1 `  F )  =  ( vol `  A ) )
 
Theoremitg1addlem1 19047* Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  C_  ( `' F " { k } )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  dom  vol )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  ( vol `  B )  e.  RR )   =>    |-  ( ph  ->  ( vol `  U_ k  e.  A  B )  = 
 sum_ k  e.  A  ( vol `  B )
 )
 
Theoremi1faddlem 19048* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ( ph  /\  A  e.  CC )  ->  ( `' ( F  o F  +  G ) " { A }
 )  =  U_ y  e.  ran  G ( ( `' F " { ( A  -  y ) }
 )  i^i  ( `' G " { y }
 ) ) )
 
Theoremi1fmullem 19049* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ( ph  /\  A  e.  ( CC  \  { 0 } )
 )  ->  ( `' ( F  o F  x.  G ) " { A } )  =  U_ y  e.  ( ran  G 
 \  { 0 } ) ( ( `' F " { ( A  /  y ) }
 )  i^i  ( `' G " { y }
 ) ) )
 
Theoremi1fadd 19050 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ph  ->  ( F  o F  +  G )  e.  dom  S.1 )
 
Theoremi1fmul 19051 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e.  dom  S.1 )
 
Theoremitg1addlem2 19052* Lemma for itg1add 19056. The function  I represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both  i and  j are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 19054 and itg1addlem5 19055. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   =>    |-  ( ph  ->  I :
 ( RR  X.  RR )
 --> RR )
 
Theoremitg1addlem3 19053* Lemma for itg1add 19056. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   =>    |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A I B )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
 
Theoremitg1addlem4 19054* Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   &    |-  P  =  (  +  |`  ( ran  F  X.  ran  G ) )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G (
 ( y  +  z
 )  x.  ( y I z ) ) )
 
Theoremitg1addlem5 19055* Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   &    |-  P  =  (  +  |`  ( ran  F  X.  ran  G ) )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  (
 ( S.1 `  F )  +  ( S.1 `  G ) ) )
 
Theoremitg1add 19056 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  (
 ( S.1 `  F )  +  ( S.1 `  G ) ) )
 
Theoremi1fmulclem 19057 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ( (
 ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( `' ( ( RR  X.  { A } )  o F  x.  F )
 " { B }
 )  =  ( `' F " { ( B  /  A ) }
 ) )
 
Theoremi1fmulc 19058 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
 
Theoremitg1mulc 19059 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 S.1 `  ( ( RR  X.  { A }
 )  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
 
Theoremi1fres 19060* The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside  A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
 |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )   =>    |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
 
Theoremi1fpos 19061* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  G  =  ( x  e.  RR  |->  if (
 0  <_  ( F `  x ) ,  ( F `  x ) ,  0 ) )   =>    |-  ( F  e.  dom  S.1  ->  G  e.  dom  S.1 )
 
Theoremi1fposd 19062* Deduction form of i1fposd 19062. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ph  ->  ( x  e.  RR  |->  A )  e.  dom  S.1 )   =>    |-  ( ph  ->  ( x  e.  RR  |->  if ( 0  <_  A ,  A ,  0 ) )  e.  dom  S.1 )
 
Theoremi1fsub 19063 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  o F  -  G )  e.  dom  S.1 )
 
Theoremitg1sub 19064 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( S.1 `  ( F  o F  -  G ) )  =  ( ( S.1 `  F )  -  ( S.1 `  G ) ) )
 
Theoremitg10a 19065* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  =  0 )   =>    |-  ( ph  ->  ( S.1 `  F )  =  0 )
 
Theoremitg1ge0a 19066* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  0  <_  ( F `  x ) )   =>    |-  ( ph  ->  0  <_  ( S.1 `  F ) )
 
Theoremitg1lea 19067* Approximate version of itg1le 19068. If  F  <_  G for almost all  x, then  S.1 F  <_  S.1 G. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  ( ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x ) 
 <_  ( G `  x ) )   =>    |-  ( ph  ->  ( S.1 `  F )  <_  ( S.1 `  G )
 )
 
Theoremitg1le 19068 If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1  /\  F  o R  <_  G )  ->  ( S.1 `  F )  <_  ( S.1 `  G ) )
 
Theoremitg1climres 19069* Restricting the simple function  F to the increasing sequence  A ( n ) of measurable sets whose union is  RR yields a sequence of simple functions whose integrals approach the integral of  F. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( ph  ->  A : NN --> dom  vol )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( A `  n )  C_  ( A `  ( n  +  1 ) ) )   &    |-  ( ph  ->  U.
 ran  A  =  RR )   &    |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  ( A `  n ) ,  ( F `  x ) ,  0 )
 )   =>    |-  ( ph  ->  ( n  e.  NN  |->  ( S.1 `  G ) )  ~~>  ( S.1 `  F ) )
 
Theoremmbfi1fseqlem1 19070* Lemma for mbfi1fseq 19076. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   =>    |-  ( ph  ->  J : ( NN  X.  RR ) --> ( 0 [,)  +oo ) )
 
Theoremmbfi1fseqlem2 19071* Lemma for mbfi1fseq 19076. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A ) ,  0 ) ) )
 
Theoremmbfi1fseqlem3 19072* Lemma for mbfi1fseq 19076. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ( ph  /\  A  e.  NN )  ->  ( G `  A ) : RR --> ran  ( m  e.  ( 0 ... ( A  x.  (
 2 ^ A ) ) )  |->  ( m 
 /  ( 2 ^ A ) ) ) )
 
Theoremmbfi1fseqlem4 19073* Lemma for mbfi1fseq 19076. This lemma is not as interesting as it is long - it is simply checking that  G is in fact a sequence of simple functions, by verifying that its range is in  ( 0 ... n 2 ^ n
)  /  ( 2 ^ n ) (which is to say, the numbers from  0 to  n in increments of  1  / 
( 2 ^ n
)), and also that the preimage of each point  k is measurable, because it is equal to  ( -u n [,] n )  i^i  ( `' F " ( k [,) k  +  1  /  ( 2 ^ n ) ) ) for  k  <  n and  ( -u n [,] n
)  i^i  ( `' F " ( k [,) 
+oo ) ) for  k  =  n. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ph  ->  G : NN --> dom  S.1 )
 
Theoremmbfi1fseqlem5 19074* Lemma for mbfi1fseq 19076. Verify that  G describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ( ph  /\  A  e.  NN )  ->  ( 0 p  o R  <_  ( G `  A )  /\  ( G `
  A )  o R  <_  ( G `  ( A  +  1 ) ) ) )
 
Theoremmbfi1fseqlem6 19075* Lemma for mbfi1fseq 19076. Verify that  G converges pointwise to  F, and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\  A. n  e.  NN  (
 0 p  o R  <_  ( g `  n )  /\  ( g `  n )  o R  <_  ( g `  ( n  +  1 )
 ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `
  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1fseq 19076* A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function  G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\  A. n  e.  NN  (
 0 p  o R  <_  ( g `  n )  /\  ( g `  n )  o R  <_  ( g `  ( n  +  1 )
 ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `
  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1flimlem 19077* Lemma for mbfi1flim 19078. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> RR )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1flim 19078* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. x  e.  A  ( n  e.  NN  |->  ( ( g `  n ) `  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfmullem2 19079* Lemma for mbfmul 19081. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( n  e.  NN  |->  ( ( P `  n ) `
  x ) )  ~~>  ( F `  x ) )   &    |-  ( ph  ->  Q : NN --> dom  S.1 )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( n  e.  NN  |->  ( ( Q `  n ) `
  x ) )  ~~>  ( G `  x ) )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremmbfmullem 19080 Lemma for mbfmul 19081. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremmbfmul 19081 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremitg2lcl 19082* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  L  C_  RR*
 
Theoremitg2val 19083* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( F : RR
 --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
 )
 
Theoremitg2l 19084* Elementhood in the set  L of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( A  e.  L 
 <-> 
 E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g )
 ) )
 
Theoremitg2lr 19085* Sufficient condition for elementhood in the set  L. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( ( G  e.  dom  S.1  /\  G  o R  <_  F ) 
 ->  ( S.1 `  G )  e.  L )
 
Theoremxrge0f 19086 A real function is a nonnegative extended real function if all its values are greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
 |-  ( ( F : RR
 --> RR  /\  0 p  o R  <_  F )  ->  F : RR --> ( 0 [,]  +oo ) )
 
Theoremitg2cl 19087 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  e.  RR* )
 
Theoremitg2ub 19088 The integral of a nonnegative real function  F is an upper bound on the integrals of all simple functions  G dominated by  F. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  G  e.  dom  S.1  /\  G  o R  <_  F )  ->  ( S.1 `  G )  <_  ( S.2 `  F ) )
 
Theoremitg2leub 19089* Any upper bound on the integrals of all simple functions  G dominated by  F is greater than  ( S.2 `  F
), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  A  e.  RR* )  ->  ( ( S.2 `  F )  <_  A  <->  A. g  e.  dom  S.1 ( g  o R  <_  F  ->  ( S.1 `  g )  <_  A ) ) )
 
Theoremitg2ge0 19090 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  0  <_  ( S.2 `  F ) )
 
Theoremitg2itg1 19091 The integral of a nonnegative simple function using  S.2 is the same as its value under  S.1. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  0 p  o R  <_  F )  ->  ( S.2 `  F )  =  ( S.1 `  F ) )
 
Theoremitg20 19092 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( S.2 `  ( RR  X.  { 0 } ) )  =  0
 
Theoremitg2lecl 19093 If an  S.2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  A  e.  RR  /\  ( S.2 `  F )  <_  A )  ->  ( S.2 `  F )  e.  RR )
 
Theoremitg2le 19094 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo )  /\  F  o R  <_  G )  ->  ( S.2 `  F )  <_  ( S.2 `  G )
 )
 
Theoremitg2const 19095* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  ( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( B  x.  ( vol `  A ) ) )
 
Theoremitg2const2 19096* When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR+ )  ->  ( ( vol `  A )  e.  RR  <->  ( S.2 `  ( x  e. 
 RR  |->  if ( x  e.  A ,  B , 
 0 ) ) )  e.  RR ) )
 
Theoremitg2seq 19097* Definitional property of the  S.2 integral: for any function  F there is a countable sequence 
g of simple functions less than  F whose integrals converge to the integral of  F. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 19110, but unlike that theorem this one doesn't require  F to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. n  e.  NN  ( g `  n )  o R  <_  F  /\  ( S.2 `  F )  =  sup ( ran  ( n  e.  NN  |->  ( S.1 `  ( g `  n ) ) ) ,  RR* ,  <  )
 ) )
 
Theoremitg2uba 19098* Approximate version of itg2ub 19088. If  F approximately dominates  G, then  S.1 G  <_  S.2 F. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( G `
  x )  <_  ( F `  x ) )   =>    |-  ( ph  ->  ( S.1 `  G )  <_  ( S.2 `  F )
 )
 
Theoremitg2lea 19099* Approximate version of itg2le 19094. If  F  <_  G for almost all  x, then  S.2 F  <_  S.2 G. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  <_  ( G `  x ) )   =>    |-  ( ph  ->  ( S.2 `  F )  <_  ( S.2 `  G )
 )
 
Theoremitg2eqa 19100* Approximate equality of integrals. If  F  =  G for almost all  x, then  S.2 F  = 
S.2 G. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  ( S.2 `  F )  =  ( S.2 `  G ) )
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