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Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmeasinb 29401* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑥𝑆 ↦ (𝑀‘(𝑥𝐴))) ∈ (measures‘𝑆))
 
Theoremmeasres 29402 Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇) ∈ (measures‘𝑇))
 
Theoremmeasinb2 29403* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑥 ∈ (𝑆 ∩ 𝒫 𝐴) ↦ (𝑀‘(𝑥𝐴))) ∈ (measures‘(𝑆 ∩ 𝒫 𝐴)))
 
TheoremmeasdivcstOLD 29404* Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ+) → (𝑥𝑆 ↦ ((𝑀𝑥) /𝑒 𝐴)) ∈ (measures‘𝑆))
 
Theoremmeasdivcst 29405 Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (Revised by Thierry Arnoux, 30-Jan-2017.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ+) → (𝑀𝑓/𝑐 /𝑒 𝐴) ∈ (measures‘𝑆))
 
20.3.15.7  The counting measure
 
Theoremcntmeas 29406 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑆 ran sigAlgebra → (# ↾ 𝑆) ∈ (measures‘𝑆))
 
Theorempwcntmeas 29407 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝑂𝑉 → (# ↾ 𝒫 𝑂) ∈ (measures‘𝒫 𝑂))
 
Theoremcntnevol 29408 Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
(# ↾ 𝒫 𝑂) ≠ vol
 
20.3.15.8  The Lebesgue measure - misc additions
 
Theoremvoliune 29409 The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for +∞ for the measure of any set in the sum. Cf. ovoliun 22958 and voliun 23007. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴))
 
Theoremvolfiniune 29410* The Lebesgue measure function is countably additive. This theorem is to volfiniun 23000 what voliune 29409 is to voliun 23007. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((𝐴 ∈ Fin ∧ ∀𝑛𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛𝐴 𝐵) → (vol‘ 𝑛𝐴 𝐵) = Σ*𝑛𝐴(vol‘𝐵))
 
Theoremvolmeas 29411 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
vol ∈ (measures‘dom vol)
 
20.3.15.9  The Dirac delta measure
 
Syntaxcdde 29412 Extend class notation to include the Dirac delta measure.
class δ
 
Definitiondf-dde 29413 Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
 
Theoremddeval1 29414 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
 
Theoremddeval0 29415 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)
 
Theoremddemeas 29416 The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
δ ∈ (measures‘𝒫 ℝ)
 
20.3.15.10  The 'almost everywhere' relation
 
Syntaxcae 29417 Extend class notation to include the 'almost everywhere' relation.
class a.e.
 
Syntaxcfae 29418 Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
 
Definitiondf-ae 29419* 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‘( dom 𝑚𝑎)) = 0}
 
Theoremrelae 29420 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Rel a.e.
 
Theorembrae 29421 'almost everywhere' relation for a measure and a measurable set 𝐴. (Contributed by Thierry Arnoux, 20-Oct-2017.)
((𝑀 ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
 
Theorembraew 29422* 'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.)
dom 𝑀 = 𝑂       (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
 
Theoremtruae 29423* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
dom 𝑀 = 𝑂    &   (𝜑𝑀 ran measures)    &   (𝜑𝜓)       (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)
 
Theoremaean 29424* A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.)
dom 𝑀 = 𝑂       ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ ({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀)))
 
Definitiondf-fae 29425* 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 ∈ V, 𝑚 ran measures ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑟𝑚 dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟𝑚 dom 𝑚)) ∧ {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚)})
 
Theoremfaeval 29426* Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.)
((𝑅 ∈ V ∧ 𝑀 ran measures) → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅𝑚 dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅𝑚 dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)})
 
Theoremrelfae 29427 The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
((𝑅 ∈ V ∧ 𝑀 ran measures) → Rel (𝑅~ a.e.𝑀))
 
Theorembrfae 29428* '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    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ (𝐷𝑚 dom 𝑀))    &   (𝜑𝐺 ∈ (𝐷𝑚 dom 𝑀))       (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
 
20.3.15.11  Measurable functions
 
Syntaxcmbfm 29429 Extend class notation with the measurable functions builder.
class MblFnM
 
Definitiondf-mbfm 29430* 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 𝑠 and 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 to , see df-mbf 23072. (Contributed by Thierry Arnoux, 23-Jan-2017.)

MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
 
Theoremismbfm 29431* The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23081. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
 
Theoremelunirnmbfm 29432* The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
 
Theoremmbfmfun 29433 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝜑𝐹 ran MblFnM)       (𝜑 → Fun 𝐹)
 
Theoremmbfmf 29434 A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))       (𝜑𝐹: 𝑆 𝑇)
 
Theoremisanmbfm 29435 The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))       (𝜑𝐹 ran MblFnM)
 
Theoremmbfmcnvima 29436 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))    &   (𝜑𝐴𝑇)       (𝜑 → (𝐹𝐴) ∈ 𝑆)
 
Theoremmbfmbfm 29437 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝜑𝑀 ran measures)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))       (𝜑𝐹 ran MblFnM)
 
Theoremmbfmcst 29438* A constant function is measurable. Cf. mbfconst 23086. (Contributed by Thierry Arnoux, 26-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 = (𝑥 𝑆𝐴))    &   (𝜑𝐴 𝑇)       (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
 
Theorem1stmbfm 29439 The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
 
Theorem2ndmbfm 29440 The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
 
Theoremimambfm 29441* 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.)
(𝜑𝐾 ∈ V)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 = (sigaGen‘𝐾))       (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹: 𝑆 𝑇 ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝑆)))
 
Theoremcnmbfm 29442 A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑆 = (sigaGen‘𝐽))    &   (𝜑𝑇 = (sigaGen‘𝐾))       (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
 
Theoremmbfmco 29443 The composition of two measurable functions is measurable. ( cf. cnmpt11 21182) (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝑅 ran sigAlgebra)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑅MblFnM𝑆))    &   (𝜑𝐺 ∈ (𝑆MblFnM𝑇))       (𝜑 → (𝐺𝐹) ∈ (𝑅MblFnM𝑇))
 
Theoremmbfmco2 29444* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21184). (Contributed by Thierry Arnoux, 6-Jun-2017.)
(𝜑𝑅 ran sigAlgebra)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑅MblFnM𝑆))    &   (𝜑𝐺 ∈ (𝑅MblFnM𝑇))    &   𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
 
Theoremmbfmvolf 29445 Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
(𝐹 ∈ (dom volMblFnM𝔅) → 𝐹:ℝ⟶ℝ)
 
Theoremelmbfmvol2 29446 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 ∈ (dom volMblFnM𝔅) → 𝐹 ∈ MblFn)
 
Theoremmbfmcnt 29447 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑𝑚 𝑂))
 
20.3.15.12  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 29448* The base set for the generator of the Borel sigma-algebra on (ℝ × ℝ) is indeed (ℝ × ℝ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
 
Theoremdya2ub 29449 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
(𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅)
 
Theoremsxbrsigalem0 29450* The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
 
Theoremsxbrsigalem3 29451* The sigma-algebra generated by the closed half-spaces of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed sets of (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
𝐽 = (topGen‘ran (,))       (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ⊆ (sigaGen‘(Clsd‘(𝐽 ×t 𝐽)))
 
Theoremdya2iocival 29452* The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23052. (Contributed by Thierry Arnoux, 24-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
 
Theoremdya2iocress 29453* Dyadic intervals are subsets of . (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ⊆ ℝ)
 
Theoremdya2iocbrsiga 29454* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅)
 
Theoremdya2icobrsiga 29455* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ran 𝐼 ⊆ 𝔅
 
Theoremdya2icoseg 29456* For any point and any closed-below, open-above interval of 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑁 = (⌊‘(1 − (2 logb 𝐷)))       ((𝑋 ∈ ℝ ∧ 𝐷 ∈ ℝ+) → ∃𝑏 ∈ ran 𝐼(𝑋𝑏𝑏 ⊆ ((𝑋𝐷)(,)(𝑋 + 𝐷))))
 
Theoremdya2icoseg2 29457* For any point and any open interval of 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋𝑏𝑏𝐸))
 
Theoremdya2iocrfn 29458* The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       𝑅 Fn (ran 𝐼 × ran 𝐼)
 
Theoremdya2iocct 29459* 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       ran 𝑅 ≼ ω
 
Theoremdya2iocnrect 29460* For any point of an open rectangle in (ℝ × ℝ), 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    &   𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
 
Theoremdya2iocnei 29461* For any point of an open set of the usual topology on (ℝ × ℝ) 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
 
Theoremdya2iocuni 29462* Every open set of (ℝ × ℝ) is a union of closed-below open-above dyadic rational rectangular subsets of (ℝ × ℝ). This union must be a countable union by dya2iocct 29459. (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (𝐴 ∈ (𝐽 ×t 𝐽) → ∃𝑐 ∈ 𝒫 ran 𝑅 𝑐 = 𝐴)
 
Theoremdya2iocucvr 29463* The dyadic rectangular set collection covers (ℝ × ℝ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))        ran 𝑅 = (ℝ × ℝ)
 
Theoremsxbrsigalem1 29464* The Borel algebra on (ℝ × ℝ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅)
 
Theoremsxbrsigalem2 29465* The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed half-spaces of (ℝ × ℝ). 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
 
Theoremsxbrsigalem4 29466* The Borel algebra on (ℝ × ℝ) is generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). 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 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅)
 
Theoremsxbrsigalem5 29467* First direction for sxbrsiga 29469. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅 ×s 𝔅)
 
Theoremsxbrsigalem6 29468 First direction for sxbrsiga 29469, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
𝐽 = (topGen‘ran (,))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅 ×s 𝔅)
 
Theoremsxbrsiga 29469 The product sigma-algebra (𝔅 ×s 𝔅) is the Borel algebra on (ℝ × ℝ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
𝐽 = (topGen‘ran (,))       (𝔅 ×s 𝔅) = (sigaGen‘(𝐽 ×t 𝐽))
 
20.3.15.13  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 29482) - it is monotone (omsmon 29483) - it is countably sub-additive (omssubadd 29485) See Definition 1.11.1 of [Bogachev] p. 41.

 
Syntaxcoms 29470 Class declaration for the outer measure construction function.
class toOMeas
 
Syntaxcomsold 29471 Class declaration for the outer measure construction function (old version).
class toOMeas
 
Definitiondf-oms 29472* Define a function constructing an outer measure. See omsval 29474 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 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
 
Definitiondf-omsOLD 29473* Define a function constructing an outer measure. See omsval 29474 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of df-oms 29472 as of 4-Oct-2020. (New usage is discouraged.)
toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ sup(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
 
Theoremomsval 29474* 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 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
 
Theoremomsfval 29475* Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
 
Theoremomscl 29476* A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)) ⊆ (0[,]+∞))
 
Theoremomsf 29477 A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 dom 𝑅⟶(0[,]+∞))
 
TheoremomsvalOLD 29478* Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsval 29474 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ sup(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
 
TheoremomsfvalOLD 29479* Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsfval 29475 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → ((toOMeas‘𝑅)‘𝐴) = sup(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
 
TheoremomsclOLD 29480* A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) Obsolete version of omscl 29476 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)) ⊆ (0[,]+∞))
 
TheoremomsfOLD 29481 A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) Obsolete version of omsf 29477 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 dom 𝑅⟶(0[,]+∞))
 
Theoremoms0 29482 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)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀‘∅) = 0)
 
Theoremomsmon 29483 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)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 𝑄)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
Theoremomssubaddlem 29484* 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)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴 𝑄)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)}Σ*𝑤𝑥(𝑅𝑤) < ((𝑀𝐴) + 𝐸))
 
Theoremomssubadd 29485* 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)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   ((𝜑𝑦𝑋) → 𝐴 𝑄)    &   (𝜑𝑋 ≼ ω)       (𝜑 → (𝑀 𝑦𝑋 𝐴) ≤ Σ*𝑦𝑋(𝑀𝐴))
 
Theoremoms0OLD 29486 A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of oms0 29482 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀‘∅) = 0)
 
TheoremomsmonOLD 29487 A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) Obsolete version of omsmon 29483 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 𝑄)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
TheoremomssubaddlemOLD 29488* 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.) Obsolete version of omssubaddlem 29484 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴 𝑄)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)}Σ*𝑤𝑥(𝑅𝑤) < ((𝑀𝐴) + 𝐸))
 
TheoremomssubaddOLD 29489* A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) Obsolete version of omssubadd 29485 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   ((𝜑𝑦𝑋) → 𝐴 𝑄)    &   (𝜑𝑋 ≼ ω)       (𝜑 → (𝑀 𝑦𝑋 𝐴) ≤ Σ*𝑦𝑋(𝑀𝐴))
 
Syntaxccarsg 29490 Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
 
Definitiondf-carsg 29491* Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 29492 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)})
 
Theoremcarsgval 29492* Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
 
Theoremcarsgcl 29493 Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
 
Theoremelcarsg 29494* Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
 
Theorembaselcarsg 29495 The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑𝑂 ∈ (toCaraSiga‘𝑀))
 
Theorem0elcarsg 29496 The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
 
Theoremcarsguni 29497 The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 (toCaraSiga‘𝑀) = 𝑂)
 
Theoremelcarsgss 29498 Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑𝐴𝑂)
 
Theoremdifelcarsg 29499 The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝑂𝐴) ∈ (toCaraSiga‘𝑀))
 
Theoreminelcarsg 29500* The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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