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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
12.2.2  Lebesgue integration
 
Syntaxcmbf 18801 Extend class notation with the class of measurable functions.
 class MblFn
 
Syntaxcitg1 18802 Extend class notation with the Lebesgue integral for simple functions.
 class  S.1
 
Syntaxcitg2 18803 Extend class notation with the Lebesgue integral for nonnegative functions.
 class  S.2
 
Syntaxcibl 18804 Extend class notation with the class of integrable functions.
 class  L ^1
 
Syntaxcitg 18805 Extend class notation with the general Lebesgue integral.
 class  S. A B  _d x
 
Syntaxcdit 18806 Extend class notation with the directed integral.
 class  S__ [ A  ->  B ] C  _d x
 
Definitiondf-mbf 18807* 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 18717) 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 18808* 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 18809* 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 18810* 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 18811* 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 18809 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 18809 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 18812 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 18813* The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 18817 and ismbfcn 18818 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 18814 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  F : dom  F --> CC )
 
Theoremmbfdm 18815 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F  e. MblFn  ->  dom 
 F  e.  dom  vol )
 
Theoremmbfconstlem 18816 Lemma for mbfconst 18822. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  C  e.  RR )  ->  ( `' ( A  X.  { C }
 ) " B )  e. 
 dom  vol )
 
Theoremismbf 18817* 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 18717. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol ) )
 
Theoremismbfcn 18818 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 18819 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 18820 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 18821 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 18822 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 18823 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( A  e.  dom  vol 
 ->  (  _I  |`  A )  e. MblFn )
 
Theoremmbfmptcl 18824* 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 18825* 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 18826* 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 18827* 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 18841. (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 18828* 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 18829* Lemma for mbfeqa 18830. (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 18830* 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 18831 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 18832 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 18833* 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 18834 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 18835 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 18836* 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 18837* 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 18838* 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 18839* Converse to mbfpos 18838. (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 18840* 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 18841* Simplified form of ismbfd 18827. (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 18842* Lemma for mbfimaopn 18843. (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 18843 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 18845, 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 18844 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 18845 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 18846 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 18847 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 18848 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 18849 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 18850* 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 18851* 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 18852* 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 18853* 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 18854* 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 18855* 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 18856 Extend class notation to include the zero polynomial.
 class 
 0 p
 
Definitiondf-0p 18857 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  0 p  =  ( CC  X.  { 0 } )
 
Theorem0pval 18858 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  CC  ->  ( 0 p `  A )  =  0
 )
 
Theorem0plef 18859 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 18860 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 18861 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 18811); 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 18862 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F  e. MblFn )
 
Theoremi1ff 18863 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 18864 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( F  e.  dom  S.1 
 ->  ran  F  e.  Fin )
 
Theoremi1fima 18865 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 18866 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 18867 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 18868* 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 18869 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 18870* 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 18871* 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 18872 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 18873 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 18874 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( RR  X.  {
 0 } )  e. 
 dom  S.1
 
Theoremitg10 18875 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( S.1 `  ( RR  X.  { 0 } ) )  =  0
 
Theoremi1f1lem 18876* Lemma for i1f1 18877 and itg11 18878. (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 18877* 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 18878* 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 18879* 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 18880* 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 18881* 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 18882 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 18883 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 18884* Lemma for itg1add 18888. 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 18886 and itg1addlem5 18887. (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 18885* Lemma for itg1add 18888. (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 18886* 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 18887* 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 18888 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 18889 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 18890 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 18891 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 18892* 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 18893* 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 18894* Deduction form of i1fposd 18894. (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 18895 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 18896 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 18897* 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 18898* 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 18899* Approximate version of itg1le 18900. 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 18900 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 ) )
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