P. 343 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.18.10  The 'almost everywhere' relation
Syntax	cae 34201	Extend class notation to include the 'almost everywhere' relation.
class a.e.
Syntax	cfae 34202	Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
Definition	df-ae 34203*	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 34204	'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ Rel a.e.
Theorem	brae 34205	'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 34206*	'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 34207*	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 34208*	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 34209*	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 34210*	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 34211	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 34212*	'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))
21.3.18.11  Measurable functions
Syntax	cmbfm 34213	Extend class notation with the measurable functions builder.
class MblFnM
Definition	df-mbfm 34214*	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 25673. (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 34215*	The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 25682. (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 34216*	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 34217	A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	mbfmf 34218	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 34219	Obsolete version of isanmbfm 34221 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 34220	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 34221	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 34222	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 34223	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 34224*	A constant function is measurable. Cf. mbfconst 25687. (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 34225	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 34226	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 34227*	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 34228	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 34229	The composition of two measurable functions is measurable. See cnmpt11 23692. (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 34230*	The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23694). (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 34231	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 34232	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 34233	All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))
21.3.18.12  Borel Algebra on ` ( RR X. RR ) `
Theorem	br2base 34234*	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 34235	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 34236*	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 34237*	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 34238*	The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25654. (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 34239*	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 34240*	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 34241*	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 34242*	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 34243*	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 34244*	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 34245*	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 34246*	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 34247*	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 34248*	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 34245. (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 34249*	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 34250*	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 34251*	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 34252*	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 34253*	First direction for sxbrsiga 34255. (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 34254	First direction for sxbrsiga 34255, 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 34255	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))
21.3.18.13  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 34262) - it is monotone (omsmon 34263) - it is countably sub-additive (omssubadd 34265) See Definition 1.11.1 of [Bogachev] p. 41.
Syntax	coms 34256	Class declaration for the outer measure construction function.
class toOMeas
Definition	df-oms 34257*	Define a function constructing an outer measure. See omsval 34258 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 34258*	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 34259*	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 34260*	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 34261	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 34262	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 34263	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 34264*	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 34265*	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 34266	Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
Definition	df-carsg 34267*	Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 34268 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 34268*	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 34269	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 34270*	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 34271	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 34272	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 34273	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 34274	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 34275	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 34276*	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 34277*	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 34278*	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 34279*	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 34280*	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 34281*	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 34282*	Lemma for carsgclctun 34286. (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 34283*	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 34284*	Lemma for carsgclctun 34286. (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 34285*	Lemma for carsgclctun 34286. (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 34286*	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 34287*	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 34288	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 34289*	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 34290*	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))
21.3.19  Integration
21.3.19.1  Lebesgue integral - misc additions
Theorem	itgeq12dv 34291*	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)
21.3.19.2  Bochner integral
Syntax	citgm 34292	Extend class notation with the (measure) Bochner integral.
class itgm
Syntax	csitm 34293	Extend class notation with the integral metric for simple functions.
class sitm
Syntax	csitg 34294	Extend class notation with the integral of simple functions.
class sitg
Definition	df-sitg 34295*	Define the integral of simple functions from a measurable space dom 𝑚 to a generic space 𝑤 equipped with the right scalar product. 𝑤 will later be required to be a Banach space. These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition. Moreover, for each 𝑥, the pre-image (◡𝑔 “ {𝑥}) is requested to be measurable, of finite measure. In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra. Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)
⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥)))))
Definition	df-sitm 34296*	Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))))
Theorem	sitgval 34297*	Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)))))
Theorem	issibf 34298*	The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
Theorem	sibf0 34299	The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ Mnd)    ⇒   ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
Theorem	sibfmbl 34300	A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))