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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmbfmcst 24601* A constant function is measurable. Cf. mbfconst 19519 (Contributed by Thierry Arnoux, 26-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  =  ( x  e.  U. S  |->  A ) )   &    |-  ( ph  ->  A  e.  U. T )   =>    |-  ( ph  ->  F  e.  ( SMblFnM T ) )
 
Theorem1stmbfm 24602 The first projection map is measurable with regard to the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( 1st  |`  ( U. S  X.  U. T ) )  e.  ( ( S ×s  T )MblFnM S ) )
 
Theorem2ndmbfm 24603 The second projection map is measurable with regard to the product sigma algebra (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   =>    |-  ( ph  ->  ( 2nd  |`  ( U. S  X.  U. T ) )  e.  ( ( S ×s  T )MblFnM T ) )
 
Theoremimambfm 24604* If the sigma-algebra in the range of a given function is generated by a collection of basic sets  K, then to check the measurability of that function, we need only consider inverse images of basic sets  a. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  =  (sigaGen `  K ) )   =>    |-  ( ph  ->  ( F  e.  ( SMblFnM T )  <->  ( F : U. S --> U. T  /\  A. a  e.  K  ( `' F " a )  e.  S ) ) )
 
Theoremcnmbfm 24605 A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
 |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  S  =  (sigaGen `  J ) )   &    |-  ( ph  ->  T  =  (sigaGen `  K ) )   =>    |-  ( ph  ->  F  e.  ( SMblFnM T ) )
 
Theoremmbfmco 24606 The composition of two measurable functions is measurable. ( cf. cnmpt11 17687) (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  R  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( RMblFnM S ) )   &    |-  ( ph  ->  G  e.  ( SMblFnM T ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  ( RMblFnM T ) )
 
Theoremmbfmco2 24607* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 17689). (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  ( ph  ->  R  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  T  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  F  e.  ( RMblFnM S ) )   &    |-  ( ph  ->  G  e.  ( RMblFnM T ) )   &    |-  H  =  ( x  e.  U. R  |->  <.
 ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ph  ->  H  e.  ( RMblFnM ( S ×s  T ) ) )
 
Theoremmbfmvolf 24608 Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  ( F  e.  ( dom  volMblFnM𝔅 )  ->  F : RR --> RR )
 
Theoremelmbfmvol2 24609 Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.)
 |-  ( F  e.  ( dom  volMblFnM𝔅 )  ->  F  e. MblFn )
 
Theoremmbfmcnt 24610 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
 |-  ( O  e.  V  ->  ( ~P OMblFnM𝔅 )  =  ( RR  ^m  O ) )
 
19.3.13.10  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 24611* The base set for the generator of the Borel sigma algebra on  ( RR  X.  RR ) is indeed  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
 |->  ( x  X.  y
 ) )  =  ( RR  X.  RR )
 
Theoremdya2ub 24612 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  ( R  e.  RR+  ->  (
 1  /  ( 2 ^ ( |_ `  (
 1  -  ( 2logb
 R ) ) ) ) )  <  R )
 
Theoremsxbrsigalem0 24613* The closed half-spaces of  ( RR  X.  RR ) cover  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
 |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR )
 )  u.  ran  (
 f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) )  =  ( RR 
 X.  RR )
 
Theoremsxbrsigalem3 24614* The sigma-algebra generated by the closed half-spaces of  ( RR  X.  RR ) is a subset of the sigma-algebra generated by the closed sets of  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  u. 
 ran  ( f  e. 
 RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
 
Theoremdya2iocival 24615* The function  I returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19484. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  (
 2 ^ N ) ) [,) ( ( X  +  1 ) 
 /  ( 2 ^ N ) ) ) )
 
Theoremdya2iocress 24616* Dyadic intervals are subsets of  RR. (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  C_  RR )
 
Theoremdya2iocbrsiga 24617* Dyadic intervals are Borel sets of 
RR. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  e. 𝔅 )
 
Theoremdya2icobrsiga 24618* Dyadic intervals are Borel sets of 
RR. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |- 
 ran  I  C_ 𝔅
 
Theoremdya2icoseg 24619* For any point and any closed below, opened above interval of  RR centered on that point, there is a closed below opened above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  N  =  ( |_ `  ( 1  -  ( 2logb D ) ) )   =>    |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D )
 ) ) )
 
Theoremdya2icoseg2 24620* For any point and any opened interval of  RR containing that point, there is a closed below opened above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   =>    |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E ) 
 ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
 
Theoremdya2iocrfn 24621* The function returning dyadic square covering for a given size has domain  ( ran  I  X.  ran  I ). (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  R  Fn  ( ran 
 I  X.  ran  I )
 
Theoremdya2iocct 24622* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |- 
 ran  R  ~<_  om
 
Theoremdya2iocnrect 24623* For any point of an opened rectangle in  ( RR  X.  RR ), there is a closed below opened above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   &    |-  B  =  ran  ( e  e.  ran  (,)
 ,  f  e.  ran  (,)  |->  ( e  X.  f
 ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
 
Theoremdya2iocnei 24624* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is a closed below opened above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  ( ( A  e.  ( J  tX  J ) 
 /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
 
Theoremdya2iocuni 24625* Every open set of  ( RR  X.  RR ) is a union of closed below opened above dyadic rational rectangular subsets of  ( RR  X.  RR ). This union must be a countable union by dya2iocct 24622. (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  ( A  e.  ( J  tX  J )  ->  E. c  e.  ~P  ran 
 R U. c  =  A )
 
Theoremdya2iocucvr 24626* The dyadic rectangular set collection covers  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |- 
 U. ran  R  =  ( RR  X.  RR )
 
Theoremsxbrsigalem1 24627* The Borel algebra on  ( RR  X.  RR ) is a subset of the sigma algebra generated by the dyadic closed below, opened above rectangular subsets of  ( RR  X.  RR ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4 (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (sigaGen `  ran  R )
 
Theoremsxbrsigalem2 24628* The sigma-algebra generated by the dyadic closed below, opened above rectangular subsets of  ( RR  X.  RR ) is a subset of the sigma algebra generated by the closed half-spaces of  ( RR  X.  RR ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ran  R ) 
 C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR )
 )  u.  ran  (
 f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) ) )
 
Theoremsxbrsigalem4 24629* The Borel algebra on  ( RR  X.  RR ) is generated by the dyadic closed below, opened above rectangular subsets of  ( RR  X.  RR ). Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed below, opened above instead of opened below, closed above in the proof as they are ultimately generated by the floor function. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ran  R )
 
Theoremsxbrsigalem5 24630* First direction for sxbrsiga 24632. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  /  ( 2 ^ n ) ) [,) (
 ( x  +  1 )  /  ( 2 ^ n ) ) ) )   &    |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I 
 |->  ( u  X.  v
 ) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (𝔅 ×s 𝔅 )
 
Theoremsxbrsigalem6 24631 First direction for sxbrsiga 24632, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (sigaGen `  ( J  tX  J ) )  C_  (𝔅 ×s 𝔅 )
 
Theoremsxbrsiga 24632 The product sigma-algebra  (𝔅 ×s 𝔅 ) is the Borel algebra on  ( RR  X.  RR ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )
 
19.3.14  Integration
 
19.3.14.1  Lebesgue integral - misc additions
 
Theoremitgeq12dv 24633* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  =  D )   =>    |-  ( ph  ->  S. A C  _d x  =  S. B D  _d x )
 
19.3.14.2  Bochner integral
 
Syntaxcitgm 24634 Extend class notation with the (measure) Lebesgue integral.
 class itgm
 
Syntaxcsitm 24635 Extend class notation with the integral metric for simple functions.
 class sitm
 
Syntaxcsitg 24636 Extend class notation with the integral of simple functions.
 class sitg
 
Definitiondf-sitg 24637* Define the integral of simple functions from a measurable space  dom  m to a generic space  w equipped with the right scalar product.  w will later be required to be a Banach space.

These simple functions are required to take finitely many different values: this is expressed by  ran  g  e.  Fin in the definition.

Moreover, for each  x, the pre-image  ( `' g " { x } ) is requested to be measurable, of finite measure.

In this definition,  (sigaGen `  ( TopOpen `  w
) ) is the Borel sigma-algebra on  w, and the functions  g range over the measurable functions over that Borel algebra.

Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)

 |- sitg  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( f  e.  { g  e.  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w )
 ) )  |  ( ran  g  e.  Fin  /\ 
 A. x  e.  ( ran  g  \  { ( 0g `  w ) }
 ) ( m `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) ) }  |->  ( w  gsumg  ( x  e.  ( ran  f  \  { ( 0g `  w ) }
 )  |->  ( ( (RRHom `  (Scalar `  w )
 ) `  ( m `  ( `' f " { x } ) ) ) ( .s `  w ) x ) ) ) ) )
 
Definitiondf-sitm 24638* Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take non-negative values in 
RR*, the range structure for this integral is  ( RR* ss  (
0 [,]  +oo ) ). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |- sitm  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( f  e.  dom  ( wsitg m ) ,  g  e.  dom  ( wsitg m )  |->  ( ( (
 RR* ss  ( 0 [,]  +oo ) )sitg m ) `  ( f  o F
 ( dist `  w )
 g ) ) ) )
 
Theoremsitgval 24639* Value of the simple function integral builder for a given space  W and measure  M. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   =>    |-  ( ph  ->  ( Wsitg M )  =  ( f  e.  {
 g  e.  ( dom 
 MMblFnM S )  |  ( ran  g  e.  Fin  /\ 
 A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) ) }  |->  ( W  gsumg  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `
  ( M `  ( `' f " { x } ) ) ) 
 .x.  x ) ) ) ) )
 
Theoremissibf 24640* The predicate " F is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   =>    |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S )  /\  ran 
 F  e.  Fin  /\  A. x  e.  ( ran 
 F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) ) )
 
Theoremsibf0 24641 The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  W  e.  Grp )   =>    |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
 
Theoremsibfmbl 24642 A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  F  e.  ( dom  MMblFnM S ) )
 
Theoremsibff 24643 A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  F : U. dom  M --> U. J )
 
Theoremsibfrn 24644 A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ran  F  e.  Fin )
 
Theoremsibfima 24645 Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ( ph  /\  A  e.  ( ran  F  \  {  .0.  } ) ) 
 ->  ( M `  ( `' F " { A } ) )  e.  ( 0 [,)  +oo ) )
 
Theoremsibfof 24646 Applying function operations on simple functions results in simple functions with regard to the the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  C  =  ( Base `  K )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  .+ 
 : ( B  X.  B ) --> C )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  K  e.  TopSp )   &    |-  ( ph  ->  J  e.  Fre )   &    |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g
 `  K ) )   =>    |-  ( ph  ->  ( F  o F  .+  G )  e.  dom  ( Ksitg M ) )
 
Theoremsitgfval 24647* Value of the Bochner integral for a simple function  F. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( ( Wsitg M ) `  F )  =  ( W  gsumg  ( x  e.  ( ran  F 
 \  {  .0.  }
 )  |->  ( ( H `
  ( M `  ( `' F " { x } ) ) ) 
 .x.  x ) ) ) )
 
Theoremsitgclg 24648* Closure of the Bochner integral on a simple functions. This version is very generic, thus the many hypothesis. See sitgclbn 24649 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  G  =  (Scalar `  W )   &    |-  D  =  ( (
 dist `  G )  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  W  e. CMnd )   &    |-  ( ph  ->  G  e.  DivRing )   &    |-  ( ph  ->  G  e. NrmRing )   &    |-  ( ph  ->  ( ZMod `  G )  e. NrmMod )   &    |-  ( ph  ->  (chr `  G )  =  0 )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  G  e. CUnifSp )   &    |-  ( ph  ->  (
 TopOpen `  G )  e. 
 Haus )   &    |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D ) )   &    |-  (
 ( ph  /\  m  e.  ( H " (
 0 [,)  +oo ) ) 
 /\  x  e.  B )  ->  ( m  .x.  x )  e.  B )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclbn 24649 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  G  =  (Scalar `  W )   &    |-  D  =  ( (
 dist `  G )  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  G  e. CUnifSp )   &    |-  ( ph  ->  ( TopOpen `  G )  e.  Haus )   &    |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
 )   &    |-  ( ph  ->  ( ZMod `  G )  e. NrmMod )   &    |-  ( ph  ->  (chr `  G )  =  0 )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclcn 24650 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  (Scalar `  W )  =fld )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclre 24651 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  (Scalar `  W )  =  (flds  RR ) )   =>    |-  ( ph  ->  ( ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgf 24652* The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  (
 ( ph  /\  f  e. 
 dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f )  e.  B )   =>    |-  ( ph  ->  ( Wsitg M ) : dom  ( Wsitg M ) --> B )
 
Theoremsitmval 24653* Value of the simple function integral metric for a given space  W and measure  M. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  D  =  ( dist `  W )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   =>    |-  ( ph  ->  ( Wsitm M )  =  ( f  e.  dom  ( Wsitg M ) ,  g  e.  dom  ( Wsitg M )  |->  ( ( (
 RR* ss  ( 0 [,]  +oo ) )sitg M ) `  ( f  o F D g ) ) ) )
 
Theoremsitmfval 24654 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  D  =  ( dist `  W )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( F ( Wsitm M ) G )  =  ( ( ( RR* ss  ( 0 [,]  +oo ) )sitg M ) `  ( F  o F D G ) ) )
 
Theoremsitmcl 24655 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
 |-  ( ph  ->  W  e.  Mnd )   &    |-  ( ph  ->  W  e.  * MetSp )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( F ( Wsitm M ) G )  e.  (
 0 [,]  +oo ) )
 
Definitiondf-itgm 24656* Define the Bochner integral as the extension by continuity of the Bochnel integral for simple functions.

Bogachev first defines 'fundamental in the mean' sequences, in definition 2.3.1 of [Bogachev] p. 116, and notes that those are actually Cauchy sequences for the pseudometric  ( wsitm m ).

He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 19508.

(Contributed by Thierry Arnoux, 13-Feb-2018.)

 |- itgm  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( ( (metUnif `  ( wsitm m ) )CnExt (UnifSt `  w ) ) `  ( wsitg m ) ) )
 
19.3.15  Probability
 
19.3.15.1  Probability Theory
 
Syntaxcprb 24657 Extend class notation to include the class of probability measures.
 class Prob
 
Definitiondf-prob 24658 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
 |- Prob  =  { p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
 
Theoremelprob 24659 The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  <->  ( P  e.  U.
 ran measures  /\  ( P `  U.
 dom  P )  =  1 ) )
 
Theoremdomprobmeas 24660 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  P  e.  (measures `  dom  P ) )
 
Theoremdomprobsiga 24661 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  dom  P  e.  U. ran sigAlgebra )
 
Theoremprobtot 24662 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  U. dom  P )  =  1 )
 
Theoremprob01 24663 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
 
Theoremprobnul 24664 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  (/) )  =  0 )
 
Theoremunveldomd 24665 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  U. dom  P  e.  dom 
 P )
 
Theoremunveldom 24666 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  U. dom  P  e.  dom  P )
 
Theoremnuleldmp 24667 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  (/)  e.  dom  P )
 
Theoremprobcun 24668* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the  sum_ construct cannot be used as it can handle infinite indexing set only if they are subsets of 
ZZ, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  ~P dom  P  /\  ( A  ~<_  om  /\ Disj  x  e.  A x ) ) 
 ->  ( P `  U. A )  = Σ* x  e.  A ( P `  x ) )
 
Theoremprobun 24669 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) ) )
 
Theoremprobdif 24670 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  ( A 
 \  B ) )  =  ( ( P `
  A )  -  ( P `  ( A  i^i  B ) ) ) )
 
Theoremprobinc 24671 A probabiliy law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  C_  B )  ->  ( P `  A )  <_  ( P `
  B ) )
 
Theoremprobdsb 24672 The probability of the complement of a set. That is, the probability that the event  A does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  ( U. dom  P  \  A ) )  =  ( 1  -  ( P `  A ) ) )
 
Theoremprobmeasd 24673 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  P  e.  U. ran measures )
 
Theoremprobvalrnd 24674 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   =>    |-  ( ph  ->  ( P `  A )  e. 
 RR )
 
Theoremprobtotrnd 24675 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( P `  U.
 dom  P )  e.  RR )
 
Theoremtotprobd 24676* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   &    |-  ( ph  ->  B  e.  ~P dom  P )   &    |-  ( ph  ->  U. B  =  U. dom  P )   &    |-  ( ph  ->  B  ~<_  om )   &    |-  ( ph  -> Disj  b  e.  B b )   =>    |-  ( ph  ->  ( P `  A )  = Σ* b  e.  B ( P `
  ( b  i^i 
 A ) ) )
 
Theoremtotprob 24677* Law of total probability (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( U. B  =  U. dom  P  /\  B  e.  ~P
 dom  P  /\  ( B  ~<_ 
 om  /\ Disj  b  e.  B b ) ) ) 
 ->  ( P `  A )  = Σ* b  e.  B ( P `  ( b  i^i  A ) ) )
 
TheoremprobfinmeasbOLD 24678* Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  ( M `  U. S ) ) )  e. Prob
 )
 
Theoremprobfinmeasb 24679 Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  ( M ` 
 U. S ) )  e. Prob )
 
Theoremprobmeasb 24680* Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `
  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) ) 
 /  ( M `  A ) ) )  e. Prob )
 
19.3.15.2  Conditional Probabilities
 
Syntaxccprob 24681 Extends class notation with the conditional probability builder.
 class cprob
 
Definitiondf-cndprob 24682* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |- cprob  =  ( p  e. Prob  |->  ( a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `  (
 a  i^i  b )
 )  /  ( p `  b ) ) ) )
 
Theoremcndprobval 24683 The value of the conditional probability , i.e. the probability for the event  A, given  B, under the probability law  P. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( P `  ( A  i^i  B ) ) 
 /  ( P `  B ) ) )
 
Theoremcndprobin 24684 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (
 (cprob `  P ) `  <. A ,  B >. )  x.  ( P `
  B ) )  =  ( P `  ( A  i^i  B ) ) )
 
Theoremcndprob01 24685 The conditional probability has values in  [ 0 ,  1 ]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  e.  (
 0 [,] 1 ) )
 
Theoremcndprobtot 24686 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. U. dom  P ,  A >. )  =  1 )
 
Theoremcndprobnul 24687 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. (/) ,  A >. )  =  0 )
 
Theoremcndprobprob 24688* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  B  e.  dom  P  /\  ( P `  B )  =/=  0 )  ->  (
 a  e.  dom  P  |->  ( (cprob `  P ) `  <. a ,  B >. ) )  e. Prob )
 
Theorembayesth 24689 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  A )  =/=  0  /\  ( P `  B )  =/=  0 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( ( (cprob `  P ) `  <. B ,  A >. )  x.  ( P `
  A ) ) 
 /  ( P `  B ) ) )
 
19.3.15.3  Real Valued Random Variables
 
Syntaxcrrv 24690 Extend class notation with the class of real valued random variables.
 class rRndVar
 
Definitiondf-rrv 24691 In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
 |- rRndVar  =  ( p  e. Prob  |->  ( dom 
 pMblFnM𝔅 ) )
 
Theoremrrvmbfm 24692 A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )
 
Theoremisrrvv 24693* Elementhood to the set of real-valued random variables with respect to the probability  P. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  ( X : U. dom  P --> RR  /\  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P ) ) )
 
Theoremrrvvf 24694 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : U. dom  P --> RR )
 
Theoremrrvfn 24695 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X  Fn  U. dom  P )
 
Theoremrrvdm 24696 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  dom 
 X  =  U. dom  P )
 
Theoremrrvrnss 24697 The range of a random variable as a subset of  RR. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ran 
 X  C_  RR )
 
Theoremrrvf2 24698 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : dom  X --> RR )
 
Theoremrrvdmss 24699 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 dom  P  C_  dom  X )
 
Theoremrrvfinvima 24700* For a real-value random variable  X, any open interval in 
RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P )
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