P. 304 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cfae 30301	Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
Definition	df-ae 30302*	Define 'almost everywhere' with regard to a measure M. A property holds almost everywhere if the measure of the set where it does not hold has measure zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- a.e. = { <. a , m >. | ( m ` ( U. dom m \ a ) ) = 0 }
Theorem	relae 30303	'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- Rel a.e.
Theorem	brae 30304	'almost everywhere' relation for a measure and a measurable set A. (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- ( ( M e. U. ran measures /\ A e. dom M ) -> ( Aa.e. M <-> ( M ` ( U. dom M \ A ) ) = 0 ) )
Theorem	braew 30305*	'almost everywhere' relation for a measure M and a property ph (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- U. dom M = O   =>    |- ( M e. U. ran measures -> ( { x e. O | ph }a.e. M <-> ( M ` { x e. O | -. ph } ) = 0 ) )
Theorem	truae 30306*	A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- U. dom M = O   &    |- ( ph -> M e. U. ran measures )   &    |- ( ph -> ps )   =>    |- ( ph -> { x e. O | ps }a.e. M )
Theorem	aean 30307*	A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.)
|- U. dom M = O   =>    |- ( ( M e. U. ran measures /\ { x e. O | -. ph } e. dom M /\ { x e. O | -. ps } e. dom M ) -> ( { x e. O | ( ph /\ ps ) }a.e. M <-> ( { x e. O | ph }a.e. M /\ { x e. O | ps }a.e. M ) ) )
Definition	df-fae 30308*	Define a builder for an 'almost everywhere' relation between functions, from relations between function values. In this definition, the range of f and g is enforced in order to ensure the resulting relation is a set. (Contributed by Thierry Arnoux, 22-Oct-2017.)
|- ~ a.e. = ( r e. _V , m e. U. ran measures |-> { <. f , g >. | ( ( f e. ( dom r ^m U. dom m ) /\ g e. ( dom r ^m U. dom m ) ) /\ { x e. U. dom m | ( f ` x ) r ( g ` x ) }a.e. m ) } )
Theorem	faeval 30309*	Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.)
|- ( ( R e. _V /\ M e. U. ran measures ) -> ( R~ a.e. M ) = { <. f , g >. | ( ( f e. ( dom R ^m U. dom M ) /\ g e. ( dom R ^m U. dom M ) ) /\ { x e. U. dom M | ( f ` x ) R ( g ` x ) }a.e. M ) } )
Theorem	relfae 30310	The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
|- ( ( R e. _V /\ M e. U. ran measures ) -> Rel ( R~ a.e. M ) )
Theorem	brfae 30311*	'almost everywhere' relation for two functions F and G with regard to the measure M. (Contributed by Thierry Arnoux, 22-Oct-2017.)
|- dom R = D   &    |- ( ph -> R e. _V )   &    |- ( ph -> M e. U. ran measures )   &    |- ( ph -> F e. ( D ^m U. dom M ) )   &    |- ( ph -> G e. ( D ^m U. dom M ) )   =>    |- ( ph -> ( F ( R~ a.e. M ) G <-> { x e. U. dom M | ( F ` x ) R ( G ` x ) }a.e. M ) )
20.3.16.11  Measurable functions
Syntax	cmbfm 30312	Extend class notation with the measurable functions builder.
class MblFnM
Definition	df-mbfm 30313*	Define the measurable function builder, which generates the set of measurable functions from a measurable space to another one. Here, the measurable spaces are given using their sigma-algebras s and t, and the spaces themselves are recovered by U. s and U. t. Note the similarities between the definition of measurable functions in measure theory, and of continuous functions in topology. This is the definition for the generic measure theory. For the specific case of functions from RR to CC, see df-mbf 23388. (Contributed by Thierry Arnoux, 23-Jan-2017.)
|- MblFnM = ( s e. U. ran sigAlgebra , t e. U. ran sigAlgebra |-> { f e. ( U. t ^m U. s ) | A. x e. t ( `' f " x ) e. s } )
Theorem	ismbfm 30314*	The predicate " F is a measurable function from the measurable space S to the measurable space T". Cf. ismbf 23397. (Contributed by Thierry Arnoux, 23-Jan-2017.)
|- ( ph -> S e. U. ran sigAlgebra )   &    |- ( ph -> T e. U. ran sigAlgebra )   =>    |- ( ph -> ( F e. ( SMblFnM T ) <-> ( F e. ( U. T ^m U. S ) /\ A. x e. T ( `' F " x ) e. S ) ) )
Theorem	elunirnmbfm 30315*	The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
|- ( F e. U. ran MblFnM <-> E. s e. U. ran sigAlgebra E. t e. U. ran sigAlgebra ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) )
Theorem	mbfmfun 30316	A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
|- ( ph -> F e. U. ran MblFnM )   =>    |- ( ph -> Fun F )
Theorem	mbfmf 30317	A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
|- ( ph -> S e. U. ran sigAlgebra )   &    |- ( ph -> T e. U. ran sigAlgebra )   &    |- ( ph -> F e. ( SMblFnM T ) )   =>    |- ( ph -> F : U. S --> U. T )
Theorem	isanmbfm 30318	The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
|- ( ph -> S e. U. ran sigAlgebra )   &    |- ( ph -> T e. U. ran sigAlgebra )   &    |- ( ph -> F e. ( SMblFnM T ) )   =>    |- ( ph -> F e. U. ran MblFnM )
Theorem	mbfmcnvima 30319	The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
|- ( ph -> S e. U. ran sigAlgebra )   &    |- ( ph -> T e. U. ran sigAlgebra )   &    |- ( ph -> F e. ( SMblFnM T ) )   &    |- ( ph -> A e. T )   =>    |- ( ph -> ( `' F " A ) e. S )
Theorem	mbfmbfm 30320	A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
|- ( ph -> M e. U. ran measures )   &    |- ( ph -> J e. Top )   &    |- ( ph -> F e. ( dom MMblFnM (sigaGen ` J ) ) )   =>    |- ( ph -> F e. U. ran MblFnM )
Theorem	mbfmcst 30321*	A constant function is measurable. Cf. mbfconst 23402. (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 ) )
Theorem	1stmbfm 30322	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 ) )
Theorem	2ndmbfm 30323	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 ) )
Theorem	imambfm 30324*	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 ) ) )
Theorem	cnmbfm 30325	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 ) )
Theorem	mbfmco 30326	The composition of two measurable functions is measurable. ( cf. cnmpt11 21466) (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 ) )
Theorem	mbfmco2 30327*	The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21468). (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 ) ) )
Theorem	mbfmvolf 30328	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 volMblFnM𝔅ℝ ) -> F : RR --> RR )
Theorem	elmbfmvol2 30329	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 volMblFnM𝔅ℝ ) -> F e. MblFn )
Theorem	mbfmcnt 30330	All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
|- ( O e. V -> ( ~P OMblFnM𝔅ℝ ) = ( RR ^m O ) )
20.3.16.12  Borel Algebra on ` ( RR X. RR ) `
Theorem	br2base 30331*	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. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( RR X. RR )
Theorem	dya2ub 30332	An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
|- ( R e. RR+ -> ( 1 / ( 2 ^ ( |_ ` ( 1 - ( 2 logb R ) ) ) ) ) < R )
Theorem	sxbrsigalem0 30333*	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 )
Theorem	sxbrsigalem3 30334*	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 ) ) )
Theorem	dya2iocival 30335*	The function I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23368. (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 ) ) ) )
Theorem	dya2iocress 30336*	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 )
Theorem	dya2iocbrsiga 30337*	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. 𝔅ℝ )
Theorem	dya2icobrsiga 30338*	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_ 𝔅ℝ
Theorem	dya2icoseg 30339*	For any point and any closed-below, open-above interval of RR centered on that point, there is a closed-below open-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 - ( 2 logb D ) ) )   =>    |- ( ( X e. RR /\ D e. RR+ ) -> E. b e. ran I ( X e. b /\ b C_ ( ( X - D ) (,) ( X + D ) ) ) )
Theorem	dya2icoseg2 30340*	For any point and any open interval of RR containing that point, there is a closed-below open-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 ) )
Theorem	dya2iocrfn 30341*	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 )
Theorem	dya2iocct 30342*	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
Theorem	dya2iocnrect 30343*	For any point of an open rectangle in ( RR X. RR ), there is a closed-below open-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 ) )
Theorem	dya2iocnei 30344*	For any point of an open set of the usual topology on ( RR X. RR ) there is a closed-below open-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 ) )
Theorem	dya2iocuni 30345*	Every open set of ( RR X. RR ) is a union of closed-below open-above dyadic rational rectangular subsets of ( RR X. RR ). This union must be a countable union by dya2iocct 30342. (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 )
Theorem	dya2iocucvr 30346*	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 )
Theorem	sxbrsigalem1 30347*	The Borel algebra on ( RR X. RR ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-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 )
Theorem	sxbrsigalem2 30348*	The sigma-algebra generated by the dyadic closed-below, open-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 ) ) ) ) )
Theorem	sxbrsigalem4 30349*	The Borel algebra on ( RR X. RR ) is generated by the dyadic closed-below, open-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, open-above instead of open-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 )
Theorem	sxbrsigalem5 30350*	First direction for sxbrsiga 30352. (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_ (𝔅ℝ ×s 𝔅ℝ )
Theorem	sxbrsigalem6 30351	First direction for sxbrsiga 30352, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
|- J = ( topGen ` ran (,) )   =>    |- (sigaGen ` ( J tX J ) ) C_ (𝔅ℝ ×s 𝔅ℝ )
Theorem	sxbrsiga 30352	The product sigma-algebra (𝔅ℝ ×s 𝔅ℝ ) 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 (,) )   =>    |- (𝔅ℝ ×s 𝔅ℝ ) = (sigaGen ` ( J tX J ) )
20.3.16.13  Caratheodory's extension theorem In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure R. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 30359) - it is monotone (omsmon 30360) - it is countably sub-additive (omssubadd 30362) See Definition 1.11.1 of [Bogachev] p. 41.
Syntax	coms 30353	Class declaration for the outer measure construction function.
class toOMeas
Definition	df-oms 30354*	Define a function constructing an outer measure. See omsval 30355 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- toOMeas = ( r e. _V |-> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) )
Theorem	omsval 30355*	Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- ( R e. _V -> (toOMeas ` R ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) )
Theorem	omsfval 30356*	Value of the outer measure evaluated for a given set A. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( (toOMeas ` R ) ` A ) = inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) )
Theorem	omscl 30357*	A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A e. ~P U. dom R ) -> ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) C_ ( 0 [,] +oo ) )
Theorem	omsf 30358	A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) ) -> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
Theorem	oms0 30359	A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- M = (toOMeas ` R )   &    |- ( ph -> Q e. V )   &    |- ( ph -> R : Q --> ( 0 [,] +oo ) )   &    |- ( ph -> (/) e. dom R )   &    |- ( ph -> ( R ` (/) ) = 0 )   =>    |- ( ph -> ( M ` (/) ) = 0 )
Theorem	omsmon 30360	A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- M = (toOMeas ` R )   &    |- ( ph -> Q e. V )   &    |- ( ph -> R : Q --> ( 0 [,] +oo ) )   &    |- ( ph -> A C_ B )   &    |- ( ph -> B C_ U. Q )   =>    |- ( ph -> ( M ` A ) <_ ( M ` B ) )
Theorem	omssubaddlem 30361*	For any small margin E, we can find a covering approaching the outer measure of a set A by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- M = (toOMeas ` R )   &    |- ( ph -> Q e. V )   &    |- ( ph -> R : Q --> ( 0 [,] +oo ) )   &    |- ( ph -> A C_ U. Q )   &    |- ( ph -> ( M ` A ) e. RR )   &    |- ( ph -> E e. RR+ )   =>    |- ( ph -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
Theorem	omssubadd 30362*	A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.)
|- M = (toOMeas ` R )   &    |- ( ph -> Q e. V )   &    |- ( ph -> R : Q --> ( 0 [,] +oo ) )   &    |- ( ( ph /\ y e. X ) -> A C_ U. Q )   &    |- ( ph -> X ~<_ om )   =>    |- ( ph -> ( M ` U_ y e. X A ) <_ Σ* y e. X ( M ` A ) )
Syntax	ccarsg 30363	Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
Definition	df-carsg 30364*	Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 30365 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
|- toCaraSiga = ( m e. _V |-> { a e. ~P U. dom m | A. e e. ~P U. dom m ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) } )
Theorem	carsgval 30365*	Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   =>    |- ( ph -> (toCaraSiga ` M ) = { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } )
Theorem	carsgcl 30366	Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   =>    |- ( ph -> (toCaraSiga ` M ) C_ ~P O )
Theorem	elcarsg 30367*	Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   =>    |- ( ph -> ( A e. (toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
Theorem	baselcarsg 30368	The universe set, O, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   =>    |- ( ph -> O e. (toCaraSiga ` M ) )
Theorem	0elcarsg 30369	The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   =>    |- ( ph -> (/) e. (toCaraSiga ` M ) )
Theorem	carsguni 30370	The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   =>    |- ( ph -> U. (toCaraSiga ` M ) = O )
Theorem	elcarsgss 30371	Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A e. (toCaraSiga ` M ) )   =>    |- ( ph -> A C_ O )
Theorem	difelcarsg 30372	The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A e. (toCaraSiga ` M ) )   =>    |- ( ph -> ( O \ A ) e. (toCaraSiga ` M ) )
Theorem	inelcarsg 30373*	The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A e. (toCaraSiga ` M ) )   &    |- ( ( ph /\ a e. ~P O /\ b e. ~P O ) -> ( M ` ( a u. b ) ) <_ ( ( M ` a ) +e ( M ` b ) ) )   &    |- ( ph -> B e. (toCaraSiga ` M ) )   =>    |- ( ph -> ( A i^i B ) e. (toCaraSiga ` M ) )
Theorem	unelcarsg 30374*	The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A e. (toCaraSiga ` M ) )   &    |- ( ( ph /\ a e. ~P O /\ b e. ~P O ) -> ( M ` ( a u. b ) ) <_ ( ( M ` a ) +e ( M ` b ) ) )   &    |- ( ph -> B e. (toCaraSiga ` M ) )   =>    |- ( ph -> ( A u. B ) e. (toCaraSiga ` M ) )
Theorem	difelcarsg2 30375*	The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A e. (toCaraSiga ` M ) )   &    |- ( ( ph /\ a e. ~P O /\ b e. ~P O ) -> ( M ` ( a u. b ) ) <_ ( ( M ` a ) +e ( M ` b ) ) )   &    |- ( ph -> B e. (toCaraSiga ` M ) )   =>    |- ( ph -> ( A \ B ) e. (toCaraSiga ` M ) )
Theorem	carsgmon 30376*	Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> A C_ B )   &    |- ( ph -> B e. ~P O )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   =>    |- ( ph -> ( M ` A ) <_ ( M ` B ) )
Theorem	carsgsigalem 30377*	Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   =>    |- ( ( ph /\ e e. ~P O /\ f e. ~P O ) -> ( M ` ( e u. f ) ) <_ ( ( M ` e ) +e ( M ` f ) ) )
Theorem	fiunelcarsg 30378*	The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A C_ (toCaraSiga ` M ) )   =>    |- ( ph -> U. A e. (toCaraSiga ` M ) )
Theorem	carsgclctunlem1 30379*	Lemma for carsgclctun 30383. (Contributed by Thierry Arnoux, 23-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A C_ (toCaraSiga ` M ) )   &    |- ( ph -> Disj y e. A y )   &    |- ( ph -> E e. ~P O )   =>    |- ( ph -> ( M ` ( E i^i U. A ) ) = Σ* y e. A ( M ` ( E i^i y ) ) )
Theorem	carsggect 30380*	The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ph -> -. (/) e. A )   &    |- ( ph -> A ~<_ om )   &    |- ( ph -> A C_ (toCaraSiga ` M ) )   &    |- ( ph -> Disj y e. A y )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   =>    |- ( ph -> Σ* z e. A ( M ` z ) <_ ( M ` U. A ) )
Theorem	carsgclctunlem2 30381*	Lemma for carsgclctun 30383. (Contributed by Thierry Arnoux, 25-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   &    |- ( ph -> Disj k e. NN A )   &    |- ( ( ph /\ k e. NN ) -> A e. (toCaraSiga ` M ) )   &    |- ( ph -> E e. ~P O )   &    |- ( ph -> ( M ` E ) =/= +oo )   =>    |- ( ph -> ( ( M ` ( E i^i U_ k e. NN A ) ) +e ( M ` ( E \ U_ k e. NN A ) ) ) <_ ( M ` E ) )
Theorem	carsgclctunlem3 30382*	Lemma for carsgclctun 30383. (Contributed by Thierry Arnoux, 24-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   &    |- ( ph -> A ~<_ om )   &    |- ( ph -> A C_ (toCaraSiga ` M ) )   &    |- ( ph -> E e. ~P O )   =>    |- ( ph -> ( ( M ` ( E i^i U. A ) ) +e ( M ` ( E \ U. A ) ) ) <_ ( M ` E ) )
Theorem	carsgclctun 30383*	The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   &    |- ( ph -> A ~<_ om )   &    |- ( ph -> A C_ (toCaraSiga ` M ) )   =>    |- ( ph -> U. A e. (toCaraSiga ` M ) )
Theorem	carsgsiga 30384*	The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.)
|- ( ph -> O e. V )   &    |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )   &    |- ( ph -> ( M ` (/) ) = 0 )   &    |- ( ( ph /\ x ~<_ om /\ x C_ ~P O ) -> ( M ` U. x ) <_ Σ* y e. x ( M ` y ) )   &    |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )   =>    |- ( ph -> (toCaraSiga ` M ) e. (sigAlgebra ` O ) )
Theorem	omsmeas 30385	The restriction of a constructed outer measure to Catatheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.)
|- M = (toOMeas ` R )   &    |- S = (toCaraSiga ` M )   &    |- ( ph -> Q e. V )   &    |- ( ph -> R : Q --> ( 0 [,] +oo ) )   &    |- ( ph -> (/) e. dom R )   &    |- ( ph -> ( R ` (/) ) = 0 )   =>    |- ( ph -> ( M |` S ) e. (measures ` S ) )
Theorem	pmeasmono 30386*	This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
|- ( ph -> P : R --> ( 0 [,] +oo ) )   &    |- ( ph -> ( P ` (/) ) = 0 )   &    |- ( ( ph /\ ( x ~<_ om /\ x C_ R /\ Disj y e. x y ) ) -> ( P ` U. x ) = Σ* y e. x ( P ` y ) )   &    |- ( ph -> A e. R )   &    |- ( ph -> B e. R )   &    |- ( ph -> ( B \ A ) e. R )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( P ` A ) <_ ( P ` B ) )
Theorem	pmeasadd 30387*	A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.)
|- ( ph -> P : R --> ( 0 [,] +oo ) )   &    |- ( ph -> ( P ` (/) ) = 0 )   &    |- ( ( ph /\ ( x ~<_ om /\ x C_ R /\ Disj y e. x y ) ) -> ( P ` U. x ) = Σ* y e. x ( P ` y ) )   &    |- Q = { s e. ~P ~P O | ( (/) e. s /\ A. x e. s A. y e. s ( ( x u. y ) e. s /\ ( x \ y ) e. s ) ) }   &    |- ( ph -> R e. Q )   &    |- ( ph -> A ~<_ om )   &    |- ( ( ph /\ k e. A ) -> B e. R )   &    |- ( ph -> Disj k e. A B )   =>    |- ( ph -> ( P ` U_ k e. A B ) = Σ* k e. A ( P ` B ) )
20.3.17  Integration
20.3.17.1  Lebesgue integral - misc additions
Theorem	itgeq12dv 30388*	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 )
20.3.17.2  Bochner integral
Syntax	citgm 30389	Extend class notation with the (measure) Bochner integral.
class itgm
Syntax	csitm 30390	Extend class notation with the integral metric for simple functions.
class sitm
Syntax	csitg 30391	Extend class notation with the integral of simple functions.
class sitg
Definition	df-sitg 30392*	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 ) ) ) ) )
Definition	df-sitm 30393*	Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in RR*, the range structure for this integral is ( RR*s ↾s ( 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*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` w ) g ) ) ) )
Theorem	sitgval 30394*	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 ) ) ) ) )
Theorem	issibf 30395*	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 ) ) ) )
Theorem	sibf0 30396	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. Mnd )   =>    |- ( ph -> ( U. dom M X. { .0. } ) e. dom ( Wsitg M ) )
Theorem	sibfmbl 30397	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 ) )
Theorem	sibff 30398	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 )
Theorem	sibfrn 30399	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 )
Theorem	sibfima 30400	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 ) )