P. 408 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	meaiininc 40701*	Measures are continuous from above: if E is a non-increasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
|- F/ n ph   &    |- ( ph -> M e. Meas )   &    |- ( ph -> N e. ZZ )   &    |- Z = ( ZZ>= ` N )   &    |- ( ph -> E : Z --> dom M )   &    |- ( ( ph /\ n e. Z ) -> ( E ` ( n + 1 ) ) C_ ( E ` n ) )   &    |- ( ph -> K e. ( ZZ>= ` N ) )   &    |- ( ph -> ( M ` ( E ` K ) ) e. RR )   &    |- S = ( n e. Z |-> ( M ` ( E ` n ) ) )   =>    |- ( ph -> S ~~> ( M ` |^|_ n e. Z ( E ` n ) ) )
Theorem	meaiininc2 40702*	Measures are continuous from above: if E is a non-increasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
|- F/ n ph   &    |- F/ k ph   &    |- ( ph -> M e. Meas )   &    |- ( ph -> N e. ZZ )   &    |- Z = ( ZZ>= ` N )   &    |- ( ph -> E : Z --> dom M )   &    |- ( ( ph /\ n e. Z ) -> ( E ` ( n + 1 ) ) C_ ( E ` n ) )   &    |- ( ph -> E. k e. Z ( M ` ( E ` k ) ) e. RR )   &    |- S = ( n e. Z |-> ( M ` ( E ` n ) ) )   =>    |- ( ph -> S ~~> ( M ` |^|_ n e. Z ( E ` n ) ) )
20.32.19.4  Outer measures and Caratheodory's construction Proofs for most of the theorems in section 113 of [Fremlin1]
Syntax	come 40703	Extend class notation with the class of outer measures.
class OutMeas
Definition	df-ome 40704*	Define the class of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- OutMeas = { x | ( ( ( ( x : dom x --> ( 0 [,] +oo ) /\ dom x = ~P U. dom x ) /\ ( x ` (/) ) = 0 ) /\ A. y e. ~P U. dom x A. z e. ~P y ( x ` z ) <_ ( x ` y ) ) /\ A. y e. ~P dom x ( y ~<_ om -> ( x ` U. y ) <_ (Σ^ ` ( x |` y ) ) ) ) }
Syntax	ccaragen 40705	Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
class CaraGen
Definition	df-caragen 40706*	Define the sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- CaraGen = ( o e. OutMeas |-> { e e. ~P U. dom o | A. a e. ~P U. dom o ( ( o ` ( a i^i e ) ) +e ( o ` ( a \ e ) ) ) = ( o ` a ) } )
Theorem	caragenval 40707*	The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( O e. OutMeas -> (CaraGen ` O ) = { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } )
Theorem	isome 40708*	Express the predicate " O is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( O e. V -> ( O e. OutMeas <-> ( ( ( ( O : dom O --> ( 0 [,] +oo ) /\ dom O = ~P U. dom O ) /\ ( O ` (/) ) = 0 ) /\ A. y e. ~P U. dom O A. z e. ~P y ( O ` z ) <_ ( O ` y ) ) /\ A. y e. ~P dom O ( y ~<_ om -> ( O ` U. y ) <_ (Σ^ ` ( O |` y ) ) ) ) ) )
Theorem	caragenel 40709*	Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> ( E e. S <-> ( E e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) ) )
Theorem	omef 40710	An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   =>    |- ( ph -> O : ~P X --> ( 0 [,] +oo ) )
Theorem	ome0 40711	The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   =>    |- ( ph -> ( O ` (/) ) = 0 )
Theorem	omessle 40712	The outer measure of a set is larger or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> B C_ X )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( O ` A ) <_ ( O ` B ) )
Theorem	omedm 40713	The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( O e. OutMeas -> dom O = ~P U. dom O )
Theorem	caragensplit 40714	If E is in the set generated by the Caratheodory's method, then it splits any set A in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set A. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- X = U. dom O   &    |- ( ph -> E e. S )   &    |- ( ph -> A C_ X )   =>    |- ( ph -> ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) = ( O ` A ) )
Theorem	caragenelss 40715	An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> A e. S )   &    |- X = U. dom O   =>    |- ( ph -> A C_ X )
Theorem	carageneld 40716*	Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- S = (CaraGen ` O )   &    |- ( ph -> E e. ~P X )   &    |- ( ( ph /\ a e. ~P X ) -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )   =>    |- ( ph -> E e. S )
Theorem	omecl 40717	The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   =>    |- ( ph -> ( O ` A ) e. ( 0 [,] +oo ) )
Theorem	caragenss 40718	The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- S = (CaraGen ` O )   =>    |- ( O e. OutMeas -> S C_ dom O )
Theorem	omeunile 40719	The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> Y C_ ~P X )   &    |- ( ph -> Y ~<_ om )   =>    |- ( ph -> ( O ` U. Y ) <_ (Σ^ ` ( O |` Y ) ) )
Theorem	caragen0 40720	The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> (/) e. S )
Theorem	omexrcl 40721	The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   =>    |- ( ph -> ( O ` A ) e. RR* )
Theorem	caragenunidm 40722	The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- S = (CaraGen ` O )   =>    |- ( ph -> X e. S )
Theorem	caragensspw 40723	The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- S = (CaraGen ` O )   =>    |- ( ph -> S C_ ~P X )
Theorem	omessre 40724	If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   &    |- ( ph -> ( O ` A ) e. RR )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> ( O ` B ) e. RR )
Theorem	caragenuni 40725	The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> U. S = U. dom O )
Theorem	caragenuncllem 40726	The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> E e. S )   &    |- ( ph -> F e. S )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   =>    |- ( ph -> ( ( O ` ( A i^i ( E u. F ) ) ) +e ( O ` ( A \ ( E u. F ) ) ) ) = ( O ` A ) )
Theorem	caragenuncl 40727	The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> E e. S )   &    |- ( ph -> F e. S )   =>    |- ( ph -> ( E u. F ) e. S )
Theorem	caragendifcl 40728	The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> E e. S )   =>    |- ( ph -> ( U. S \ E ) e. S )
Theorem	caragenfiiuncl 40729*	The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ k ph   &    |- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   =>    |- ( ph -> U_ k e. A B e. S )
Theorem	omeunle 40730	The outer measure of the union of two sets is less or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   &    |- ( ph -> B C_ X )   =>    |- ( ph -> ( O ` ( A u. B ) ) <_ ( ( O ` A ) +e ( O ` B ) ) )
Theorem	omeiunle 40731*	The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ n ph   &    |- F/_ n E   &    |- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- Z = ( ZZ>= ` N )   &    |- ( ph -> E : Z --> ~P X )   =>    |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) <_ (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) )
Theorem	omelesplit 40732	The outer measure of a set A is less than or equal to the extended addition of the outer measures of the decomposition induced on A by any E. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   =>    |- ( ph -> ( O ` A ) <_ ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) )
Theorem	omeiunltfirp 40733*	If the outer measure of a countable union is not +oo, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- Z = ( ZZ>= ` N )   &    |- ( ph -> E : Z --> ~P X )   &    |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) e. RR )   &    |- ( ph -> Y e. RR+ )   =>    |- ( ph -> E. z e. ( ~P Z i^i Fin ) ( O ` U_ n e. Z ( E ` n ) ) < ( sum_ n e. z ( O ` ( E ` n ) ) + Y ) )
Theorem	omeiunlempt 40734*	The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ n ph   &    |- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- Z = ( ZZ>= ` N )   &    |- ( ( ph /\ n e. Z ) -> E C_ X )   =>    |- ( ph -> ( O ` U_ n e. Z E ) <_ (Σ^ ` ( n e. Z |-> ( O ` E ) ) ) )
Theorem	carageniuncllem1 40735*	The outer measure of A i^i ( G ` n ) is the sum of the outer measures of A i^i ( F ` m ). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   &    |- ( ph -> ( O ` A ) e. RR )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> E : Z --> S )   &    |- G = ( n e. Z |-> U_ i e. ( M ... n ) ( E ` i ) )   &    |- F = ( n e. Z |-> ( ( E ` n ) \ U_ i e. ( M..^ n ) ( E ` i ) ) )   &    |- ( ph -> K e. Z )   =>    |- ( ph -> sum_ n e. ( M ... K ) ( O ` ( A i^i ( F ` n ) ) ) = ( O ` ( A i^i ( G ` K ) ) ) )
Theorem	carageniuncllem2 40736*	The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   &    |- ( ph -> ( O ` A ) e. RR )   &    |- ( ph -> M e. ZZ )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> E : Z --> S )   &    |- ( ph -> Y e. RR+ )   &    |- G = ( n e. Z |-> U_ i e. ( M ... n ) ( E ` i ) )   &    |- F = ( n e. Z |-> ( ( E ` n ) \ U_ i e. ( M..^ n ) ( E ` i ) ) )   =>    |- ( ph -> ( ( O ` ( A i^i U_ n e. Z ( E ` n ) ) ) +e ( O ` ( A \ U_ n e. Z ( E ` n ) ) ) ) <_ ( ( O ` A ) + Y ) )
Theorem	carageniuncl 40737*	The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> M e. ZZ )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> E : Z --> S )   =>    |- ( ph -> U_ n e. Z ( E ` n ) e. S )
Theorem	caragenunicl 40738	The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- ( ph -> X C_ S )   &    |- ( ph -> X ~<_ om )   =>    |- ( ph -> U. X e. S )
Theorem	caragensal 40739	Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> S e. SAlg )
Theorem	caratheodorylem1 40740*	Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> E : Z --> S )   &    |- ( ph -> Disj n e. Z ( E ` n ) )   &    |- G = ( n e. Z |-> U_ i e. ( M ... n ) ( E ` i ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( O ` ( G ` N ) ) = (Σ^ ` ( n e. ( M ... N ) |-> ( O ` ( E ` n ) ) ) ) )
Theorem	caratheodorylem2 40741*	Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- S = (CaraGen ` O )   &    |- ( ph -> E : NN --> S )   &    |- ( ph -> Disj n e. NN ( E ` n ) )   &    |- G = ( k e. NN |-> U_ n e. ( 1 ... k ) ( E ` n ) )   =>    |- ( ph -> ( O ` U_ n e. NN ( E ` n ) ) = (Σ^ ` ( n e. NN |-> ( O ` ( E ` n ) ) ) ) )
Theorem	caratheodory 40742	Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> O e. OutMeas )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> ( O |` S ) e. Meas )
Theorem	0ome 40743*	The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. V )   &    |- O = ( x e. ~P X |-> 0 )   =>    |- ( ph -> O e. OutMeas )
Theorem	isomenndlem 40744*	O is sub-additive w.r.t. countable indexed union, implies that O is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> O : ~P X --> ( 0 [,] +oo ) )   &    |- ( ph -> ( O ` (/) ) = 0 )   &    |- ( ph -> Y C_ ~P X )   &    |- ( ( ph /\ a : NN --> ~P X ) -> ( O ` U_ n e. NN ( a ` n ) ) <_ (Σ^ ` ( n e. NN |-> ( O ` ( a ` n ) ) ) ) )   &    |- ( ph -> B C_ NN )   &    |- ( ph -> F : B -1-1-onto-> Y )   &    |- A = ( n e. NN |-> if ( n e. B , ( F ` n ) , (/) ) )   =>    |- ( ph -> ( O ` U. Y ) <_ (Σ^ ` ( O |` Y ) ) )
Theorem	isomennd 40745*	Sufficient condition to prove that O is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. V )   &    |- ( ph -> O : ~P X --> ( 0 [,] +oo ) )   &    |- ( ph -> ( O ` (/) ) = 0 )   &    |- ( ( ph /\ x C_ X /\ y C_ x ) -> ( O ` y ) <_ ( O ` x ) )   &    |- ( ( ph /\ a : NN --> ~P X ) -> ( O ` U_ n e. NN ( a ` n ) ) <_ (Σ^ ` ( n e. NN |-> ( O ` ( a ` n ) ) ) ) )   =>    |- ( ph -> O e. OutMeas )
Theorem	caragenel2d 40746*	Membership in the Caratheodory's construction. Similar to carageneld 40716, but here "less then or equal" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- S = (CaraGen ` O )   &    |- ( ph -> E e. ~P X )   &    |- ( ( ph /\ a e. ~P X ) -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) <_ ( O ` a ) )   =>    |- ( ph -> E e. S )
Theorem	omege0 40747	If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   =>    |- ( ph -> 0 <_ ( O ` A ) )
Theorem	omess0 40748	If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> A C_ X )   &    |- ( ph -> ( O ` A ) = 0 )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> ( O ` B ) = 0 )
Theorem	caragencmpl 40749	A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> O e. OutMeas )   &    |- X = U. dom O   &    |- ( ph -> E C_ X )   &    |- ( ph -> ( O ` E ) = 0 )   &    |- S = (CaraGen ` O )   =>    |- ( ph -> E e. S )
20.32.19.5  Lebesgue measure on n-dimensional Real numbers Proofs for most of the theorems in section 115 of [Fremlin1]
Syntax	covoln 40750	Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
class voln*
Definition	df-ovoln 40751*	Define the outer measure for the space of multidimensional real numbers. The cardinality of x is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- voln* = ( x e. Fin |-> ( y e. ~P ( RR ^m x ) |-> if ( x = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m x ) ^m NN ) ( y C_ U_ j e. NN X_ k e. x ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. x ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) ) )
Syntax	cvoln 40752	Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
class voln
Definition	df-voln 40753	Define the Lebesgue measure for the space of multidimensional real numbers. The cardinality of x is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- voln = ( x e. Fin |-> ( (voln* ` x ) |` (CaraGen ` (voln* ` x ) ) ) )
Theorem	vonval 40754	Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> (voln ` X ) = ( (voln* ` X ) |` (CaraGen ` (voln* ` X ) ) ) )
Theorem	ovnval 40755*	Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> (voln* ` X ) = ( y e. ~P ( RR ^m X ) |-> if ( X = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) ) )
Theorem	elhoi 40756*	Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. V )   =>    |- ( ph -> ( Y e. ( ( A [,) B ) ^m X ) <-> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) ) )
Theorem	icoresmbl 40757	A closed-below, open-above real interval is measurable, when the bounds are real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ran ( [,) |` ( RR X. RR ) ) C_ dom vol
Theorem	hoissre 40758*	The projection of a half-open interval onto a single dimension is a subset of RR. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> I : X --> ( RR X. RR ) )   =>    |- ( ( ph /\ k e. X ) -> ( ( [,) o. I ) ` k ) C_ RR )
Theorem	ovnval2 40759*	Value of the Lebesgue outer measure of a subset A of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> ( (voln* ` X ) ` A ) = if ( X = (/) , 0 , inf ( M , RR* , < ) ) )
Theorem	volicorecl 40760	The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A [,) B ) ) e. RR )
Theorem	hoiprodcl 40761*	The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- F/ k ph   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I : X --> ( RR X. RR ) )   =>    |- ( ph -> prod_ k e. X ( vol ` ( ( [,) o. I ) ` k ) ) e. ( 0 [,) +oo ) )
Theorem	hoicvr 40762*	I is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- I = ( j e. NN |-> ( x e. X |-> <. -u j , j >. ) )   &    |- ( ph -> X e. Fin )   =>    |- ( ph -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) )
Theorem	hoissrrn 40763*	A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> I : X --> ( RR X. RR ) )   =>    |- ( ph -> X_ k e. X ( ( [,) o. I ) ` k ) C_ ( RR ^m X ) )
Theorem	ovn0val 40764	The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A C_ ( RR ^m (/) ) )   =>    |- ( ph -> ( (voln* ` (/) ) ` A ) = 0 )
Theorem	ovnn0val 40765*	The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> ( (voln* ` X ) ` A ) = inf ( M , RR* , < ) )
Theorem	ovnval2b 40766*	Value of the Lebesgue outer measure of a subset A of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- L = ( a e. ~P ( RR ^m X ) |-> { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } )   =>    |- ( ph -> ( (voln* ` X ) ` A ) = if ( X = (/) , 0 , inf ( ( L ` A ) , RR* , < ) ) )
Theorem	volicorescl 40767	The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( A e. ran ( [,) |` ( RR X. RR ) ) -> ( vol ` A ) e. RR )
Theorem	ovnprodcl 40768*	The product used in the definition of the outer Lebesgue measure in R^n is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- F/ k ph   &    |- ( ph -> X e. Fin )   &    |- ( ph -> F : NN --> ( ( RR X. RR ) ^m X ) )   &    |- ( ph -> I e. NN )   =>    |- ( ph -> prod_ k e. X ( vol ` ( ( [,) o. ( F ` I ) ) ` k ) ) e. ( 0 [,) +oo ) )
Theorem	hoiprodcl2 40769*	The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- F/ k ph   &    |- ( ph -> X e. Fin )   &    |- L = ( i e. ( ( RR X. RR ) ^m X ) |-> prod_ k e. X ( vol ` ( ( [,) o. i ) ` k ) ) )   &    |- ( ph -> I : X --> ( RR X. RR ) )   =>    |- ( ph -> ( L ` I ) e. ( 0 [,) +oo ) )
Theorem	hoicvrrex 40770*	Any subset of the multidimensional reals can be covered by a countable set of half-open intervals, see Definition 115A (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> Y C_ ( RR ^m X ) )   =>    |- ( ph -> E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( Y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ +oo = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) )
Theorem	ovnsupge0 40771*	The set used in the definition of the Lebesgue outer measure is a subset of the nonnegative extended reals. This is a substep for (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> M C_ ( 0 [,] +oo ) )
Theorem	ovnlecvr 40772*	Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with X the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- L = ( i e. ( ( RR X. RR ) ^m X ) |-> prod_ k e. X ( vol ` ( ( [,) o. i ) ` k ) ) )   &    |- ( ph -> I : NN --> ( ( RR X. RR ) ^m X ) )   &    |- ( ph -> A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( I ` j ) ) ` k ) )   =>    |- ( ph -> ( (voln* ` X ) ` A ) <_ (Σ^ ` ( j e. NN |-> ( L ` ( I ` j ) ) ) ) )
Theorem	ovnpnfelsup 40773*	+oo is an element of the set used in the definition of the Lebesgue outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> +oo e. M )
Theorem	ovnsslelem 40774*	The (multidimensional, nonzero-dimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A C_ B )   &    |- ( ph -> B C_ ( RR ^m X ) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   &    |- N = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( B C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> ( (voln* ` X ) ` A ) <_ ( (voln* ` X ) ` B ) )
Theorem	ovnssle 40775	The (multidimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ B )   &    |- ( ph -> B C_ ( RR ^m X ) )   =>    |- ( ph -> ( (voln* ` X ) ` A ) <_ ( (voln* ` X ) ` B ) )
Theorem	ovnlerp 40776*	The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- ( ph -> E e. RR+ )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }   =>    |- ( ph -> E. z e. M z <_ ( ( (voln* ` X ) ` A ) +e E ) )
Theorem	ovnf 40777	The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> (voln* ` X ) : ~P ( RR ^m X ) --> ( 0 [,] +oo ) )
Theorem	ovncvrrp 40778*	The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A C_ ( RR ^m X ) )   &    |- ( ph -> E e. RR+ )   &    |- C = ( a e. ~P ( RR ^m X ) |-> { l e. ( ( ( RR X. RR ) ^m X ) ^m NN ) | a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( l ` j ) ) ` k ) } )   &    |- L = ( h e. ( ( RR X. RR ) ^m X ) |-> prod_ k e. X ( vol ` ( ( [,) o. h ) ` k ) ) )   &    |- D = ( a e. ~P ( RR ^m X ) |-> ( e e. RR+ |-> { i e. ( C ` a ) | (Σ^ ` ( j e. NN |-> ( L ` ( i ` j ) ) ) ) <_ ( ( (voln* ` X ) ` a ) +e e ) } ) )   =>    |- ( ph -> E. i i e. ( ( D ` A ) ` E ) )
Theorem	ovn0lem 40779*	For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (a)(ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) }   &    |- ( ph -> inf ( M , RR* , < ) e. ( 0 [,] +oo ) )   &    |- I = ( j e. NN |-> ( l e. X |-> <. 1 , 0 >. ) )   =>    |- ( ph -> inf ( M , RR* , < ) = 0 )
Theorem	ovn0 40780	For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> ( (voln* ` X ) ` (/) ) = 0 )
Theorem	ovncl 40781	The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   =>    |- ( ph -> ( (voln* ` X ) ` A ) e. ( 0 [,] +oo ) )
Theorem	ovn02 40782	For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- (voln* ` (/) ) = ( x e. ~P { (/) } |-> 0 )
Theorem	ovnxrcl 40783	The Lebesgue outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A C_ ( RR ^m X ) )   =>    |- ( ph -> ( (voln* ` X ) ` A ) e. RR* )
Theorem	ovnsubaddlem1 40784*	The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A : NN --> ~P ( RR ^m X ) )   &    |- ( ph -> E e. RR+ )   &    |- Z = ( a e. ~P ( RR ^m X ) |-> { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } )   &    |- C = ( a e. ~P ( RR ^m X ) |-> { h e. ( ( ( RR X. RR ) ^m X ) ^m NN ) | a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( h ` j ) ) ` k ) } )   &    |- L = ( i e. ( ( RR X. RR ) ^m X ) |-> prod_ k e. X ( vol ` ( ( [,) o. i ) ` k ) ) )   &    |- D = ( a e. ~P ( RR ^m X ) |-> ( e e. RR+ |-> { i e. ( C ` a ) | (Σ^ ` ( j e. NN |-> ( L ` ( i ` j ) ) ) ) <_ ( ( (voln* ` X ) ` a ) +e e ) } ) )   &    |- ( ( ph /\ n e. NN ) -> ( I ` n ) e. ( ( D ` ( A ` n ) ) ` ( E / ( 2 ^ n ) ) ) )   &    |- ( ph -> F : NN -1-1-onto-> ( NN X. NN ) )   &    |- G = ( m e. NN |-> ( ( I ` ( 1st ` ( F ` m ) ) ) ` ( 2nd ` ( F ` m ) ) ) )   =>    |- ( ph -> ( (voln* ` X ) ` U_ n e. NN ( A ` n ) ) <_ ( (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( A ` n ) ) ) ) +e E ) )
Theorem	ovnsubaddlem2 40785*	(voln* ` X ) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A : NN --> ~P ( RR ^m X ) )   &    |- ( ph -> E e. RR+ )   &    |- Z = ( a e. ~P ( RR ^m X ) |-> { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } )   &    |- C = ( a e. ~P ( RR ^m X ) |-> { l e. ( ( ( RR X. RR ) ^m X ) ^m NN ) | a C_ U_ j e. NN X_ k e. X ( ( [,) o. ( l ` j ) ) ` k ) } )   &    |- L = ( h e. ( ( RR X. RR ) ^m X ) |-> prod_ k e. X ( vol ` ( ( [,) o. h ) ` k ) ) )   &    |- D = ( a e. ~P ( RR ^m X ) |-> ( e e. RR+ |-> { i e. ( C ` a ) | (Σ^ ` ( j e. NN |-> ( L ` ( i ` j ) ) ) ) <_ ( ( (voln* ` X ) ` a ) +e e ) } ) )   =>    |- ( ph -> ( (voln* ` X ) ` U_ n e. NN ( A ` n ) ) <_ ( (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( A ` n ) ) ) ) +e E ) )
Theorem	ovnsubadd 40786*	(voln* ` X ) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   &    |- ( ph -> A : NN --> ~P ( RR ^m X ) )   =>    |- ( ph -> ( (voln* ` X ) ` U_ n e. NN ( A ` n ) ) <_ (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( A ` n ) ) ) ) )
Theorem	ovnome 40787	(voln* ` X ) is an outer measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set X. Proposition 115D (a) of [Fremlin1] p. 30 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> (voln* ` X ) e. OutMeas )
Theorem	vonmea 40788	(voln ` X ) is a measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set X. Comments in Definition 115E of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> X e. Fin )   =>    |- ( ph -> (voln ` X ) e. Meas )
Theorem	volicon0 40789	The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( vol ` ( A [,) B ) ) = ( B - A ) )
Theorem	hsphoif 40790*	H is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- H = ( x e. RR |-> ( a e. ( RR ^m X ) |-> ( j e. X |-> if ( j e. Y , ( a ` j ) , if ( ( a ` j ) <_ x , ( a ` j ) , x ) ) ) ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> X e. V )   &    |- ( ph -> B : X --> RR )   =>    |- ( ph -> ( ( H ` A ) ` B ) : X --> RR )
Theorem	hoidmvval 40791*	The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> A : X --> RR )   &    |- ( ph -> B : X --> RR )   &    |- ( ph -> X e. Fin )   =>    |- ( ph -> ( A ( L ` X ) B ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) )
Theorem	hoissrrn2 40792*	A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- F/ k ph   &    |- ( ( ph /\ k e. X ) -> A e. RR )   &    |- ( ( ph /\ k e. X ) -> B e. RR* )   =>    |- ( ph -> X_ k e. X ( A [,) B ) C_ ( RR ^m X ) )
Theorem	hsphoival 40793*	H is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- H = ( x e. RR |-> ( a e. ( RR ^m X ) |-> ( j e. X |-> if ( j e. Y , ( a ` j ) , if ( ( a ` j ) <_ x , ( a ` j ) , x ) ) ) ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> X e. V )   &    |- ( ph -> B : X --> RR )   &    |- ( ph -> K e. X )   =>    |- ( ph -> ( ( ( H ` A ) ` B ) ` K ) = if ( K e. Y , ( B ` K ) , if ( ( B ` K ) <_ A , ( B ` K ) , A ) ) )
Theorem	hoiprodcl3 40794*	The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- F/ k ph   &    |- ( ph -> X e. Fin )   &    |- ( ( ph /\ k e. X ) -> A e. RR )   &    |- ( ( ph /\ k e. X ) -> B e. RR )   =>    |- ( ph -> prod_ k e. X ( vol ` ( A [,) B ) ) e. ( 0 [,) +oo ) )
Theorem	volicore 40795	The Lebesgue measure of a left-closed right-open interval is a real number. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A [,) B ) ) e. RR )
Theorem	hoidmvcl 40796*	The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> A : X --> RR )   &    |- ( ph -> B : X --> RR )   =>    |- ( ph -> ( A ( L ` X ) B ) e. ( 0 [,) +oo ) )
Theorem	hoidmv0val 40797*	The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> A : (/) --> RR )   &    |- ( ph -> B : (/) --> RR )   =>    |- ( ph -> ( A ( L ` (/) ) B ) = 0 )
Theorem	hoidmvn0val 40798*	The dimensional volume of a non 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> X =/= (/) )   &    |- ( ph -> A : X --> RR )   &    |- ( ph -> B : X --> RR )   =>    |- ( ph -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) )
Theorem	hsphoidmvle2 40799*	The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> Z e. ( X \ Y ) )   &    |- X = ( Y u. { Z } )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> C <_ D )   &    |- H = ( x e. RR |-> ( c e. ( RR ^m X ) |-> ( j e. X |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) )   &    |- ( ph -> A : X --> RR )   &    |- ( ph -> B : X --> RR )   =>    |- ( ph -> ( A ( L ` X ) ( ( H ` C ) ` B ) ) <_ ( A ( L ` X ) ( ( H ` D ) ` B ) ) )
Theorem	hsphoidmvle 40800*	The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> Z e. ( X \ Y ) )   &    |- X = ( Y u. { Z } )   &    |- ( ph -> C e. RR )   &    |- H = ( x e. RR |-> ( c e. ( RR ^m X ) |-> ( j e. X |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) )   &    |- ( ph -> A : X --> RR )   &    |- ( ph -> B : X --> RR )   =>    |- ( ph -> ( A ( L ` X ) ( ( H ` C ) ` B ) ) <_ ( A ( L ` X ) B ) )