P. 340 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.17.8  The Lebesgue measure - misc additions
Theorem	voliune 33901	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 25447 and voliun 25496. (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))
Theorem	volfiniune 33902*	The Lebesgue measure function is countably additive. This theorem is to volfiniun 25489 what voliune 33901 is to voliun 25496. (Contributed by Thierry Arnoux, 16-Oct-2017.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ*𝑛 ∈ 𝐴(vol‘𝐵))
Theorem	volmeas 33903	The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
⊢ vol ∈ (measures‘dom vol)
21.3.17.9  The Dirac delta measure
Syntax	cdde 33904	Extend class notation to include the Dirac delta measure.
class δ
Definition	df-dde 33905	Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
Theorem	ddeval1 33906	Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
Theorem	ddeval0 33907	Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)
Theorem	ddemeas 33908	The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ δ ∈ (measures‘𝒫 ℝ)
21.3.17.10  The 'almost everywhere' relation
Syntax	cae 33909	Extend class notation to include the 'almost everywhere' relation.
class a.e.
Syntax	cfae 33910	Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
Definition	df-ae 33911*	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 𝑚 ∖ 𝑎)) = 0}
Theorem	relae 33912	'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ Rel a.e.
Theorem	brae 33913	'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 𝑀 ∖ 𝐴)) = 0))
Theorem	braew 33914*	'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ ∪ dom 𝑀 = 𝑂    ⇒   ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
Theorem	truae 33915*	A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ ∪ dom 𝑀 = 𝑂    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)
Theorem	aean 33916*	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 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)))
Definition	df-fae 33917*	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 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)})
Theorem	faeval 33918*	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 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)})
Theorem	relfae 33919	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))
Theorem	brfae 33920*	'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    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀))    ⇒   ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))
21.3.17.11  Measurable functions
Syntax	cmbfm 33921	Extend class notation with the measurable functions builder.
class MblFnM
Definition	df-mbfm 33922*	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 ℝ to ℂ, see df-mbf 25561. (Contributed by Thierry Arnoux, 23-Jan-2017.)
⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
Theorem	ismbfm 33923*	The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 25570. (Contributed by Thierry Arnoux, 23-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))
Theorem	elunirnmbfm 33924*	The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
Theorem	mbfmfun 33925	A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	mbfmf 33926	A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    ⇒   ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
Theorem	isanmbfmOLD 33927	Obsolete version of isanmbfm 33929 as of 13-Jan-2025. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmcnvima 33928	The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    ⇒   ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆)
Theorem	isanmbfm 33929	The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmbfmOLD 33930	A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmbfm 33931	A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmcst 33932*	A constant function is measurable. Cf. mbfconst 25575. (Contributed by Thierry Arnoux, 26-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴))    &   ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑇)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
Theorem	1stmbfm 33933	The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    ⇒   ⊢ (𝜑 → (1st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
Theorem	2ndmbfm 33934	The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    ⇒   ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Theorem	imambfm 33935*	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.)
⊢ (𝜑 → 𝐾 ∈ V)    &   ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)))
Theorem	cnmbfm 33936	A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑆 = (sigaGen‘𝐽))    &   ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
Theorem	mbfmco 33937	The composition of two measurable functions is measurable. See cnmpt11 23580. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))
Theorem	mbfmco2 33938*	The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23582). (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇))    &   ⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Theorem	mbfmvolf 33939	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 volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ)
Theorem	elmbfmvol2 33940	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 volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn)
Theorem	mbfmcnt 33941	All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))
21.3.17.12  Borel Algebra on ` ( RR X. RR ) `
Theorem	br2base 33942*	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 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Theorem	dya2ub 33943	An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ (𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅)
Theorem	sxbrsigalem0 33944*	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)
Theorem	sxbrsigalem3 33945*	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)))
Theorem	dya2iocival 33946*	The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25542. (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))))
Theorem	dya2iocress 33947*	Dyadic intervals are subsets of ℝ. (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)
Theorem	dya2iocbrsiga 33948*	Dyadic intervals are Borel sets of ℝ. (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) ∈ 𝔅ℝ)
Theorem	dya2icobrsiga 33949*	Dyadic intervals are Borel sets of ℝ. (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 ⊆ 𝔅ℝ
Theorem	dya2icoseg 33950*	For any point and any closed-below, open-above interval of ℝ 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(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝐷)(,)(𝑋 + 𝐷))))
Theorem	dya2icoseg2 33951*	For any point and any open interval of ℝ 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 (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
Theorem	dya2iocrfn 33952*	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)
Theorem	dya2iocct 33953*	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
Theorem	dya2iocnrect 33954*	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) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
Theorem	dya2iocnei 33955*	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) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
Theorem	dya2iocuni 33956*	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 33953. (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)
Theorem	dya2iocucvr 33957*	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)
Theorem	sxbrsigalem1 33958*	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)
Theorem	sxbrsigalem2 33959*	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[,)+∞)))))
Theorem	sxbrsigalem4 33960*	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)
Theorem	sxbrsigalem5 33961*	First direction for sxbrsiga 33963. (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)) ⊆ (𝔅ℝ ×s 𝔅ℝ)
Theorem	sxbrsigalem6 33962	First direction for sxbrsiga 33963, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ)
Theorem	sxbrsiga 33963	The product sigma-algebra (𝔅ℝ ×s 𝔅ℝ) 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 (,))    ⇒   ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘(𝐽 ×t 𝐽))
21.3.17.13  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 33970) - it is monotone (omsmon 33971) - it is countably sub-additive (omssubadd 33973) See Definition 1.11.1 of [Bogachev] p. 41.
Syntax	coms 33964	Class declaration for the outer measure construction function.
class toOMeas
Definition	df-oms 33965*	Define a function constructing an outer measure. See omsval 33966 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 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )))
Theorem	omsval 33966*	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 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))
Theorem	omsfval 33967*	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 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))
Theorem	omscl 33968*	A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞))
Theorem	omsf 33969	A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 ∪ dom 𝑅⟶(0[,]+∞))
Theorem	oms0 33970	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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → ∅ ∈ dom 𝑅)    &   ⊢ (𝜑 → (𝑅‘∅) = 0)    ⇒   ⊢ (𝜑 → (𝑀‘∅) = 0)
Theorem	omsmon 33971	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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑄)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	omssubaddlem 33972*	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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑄)    &   ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑤 ∈ 𝑥(𝑅‘𝑤) < ((𝑀‘𝐴) + 𝐸))
Theorem	omssubadd 33973*	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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄)    &   ⊢ (𝜑 → 𝑋 ≼ ω)    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴))
Syntax	ccarsg 33974	Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
Definition	df-carsg 33975*	Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 33976 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‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
Theorem	carsgval 33976*	Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})
Theorem	carsgcl 33977	Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
Theorem	elcarsg 33978*	Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))))
Theorem	baselcarsg 33979	The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
Theorem	0elcarsg 33980	The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
Theorem	carsguni 33981	The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂)
Theorem	elcarsgss 33982	Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)
Theorem	difelcarsg 33983	The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀))
Theorem	inelcarsg 33984*	The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	unelcarsg 33985*	The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	difelcarsg2 33986*	The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	carsgmon 33987*	Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	carsgsigalem 33988*	Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓)))
Theorem	fiunelcarsg 33989*	The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgclctunlem1 33990*	Lemma for carsgclctun 33994. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
Theorem	carsggect 33991*	The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
Theorem	carsgclctunlem2 33992*	Lemma for carsgclctun 33994. (Contributed by Thierry Arnoux, 25-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    &   ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctunlem3 33993*	Lemma for carsgclctun 33994. (Contributed by Thierry Arnoux, 24-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctun 33994*	The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgsiga 33995*	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.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))
Theorem	omsmeas 33996	The restriction of a constructed outer measure to Caratheodory 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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → ∅ ∈ dom 𝑅)    &   ⊢ (𝜑 → (𝑅‘∅) = 0)    ⇒   ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆))
Theorem	pmeasmono 33997*	This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
Theorem	pmeasadd 33998*	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.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑅 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵))
21.3.18  Integration
21.3.18.1  Lebesgue integral - misc additions
Theorem	itgeq12dv 33999*	Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
21.3.18.2  Bochner integral
Syntax	citgm 34000	Extend class notation with the (measure) Bochner integral.
class itgm