P. 309 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sxuni 30801	The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
Theorem	elsx 30802	The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
20.3.16.6  Measures
Syntax	cmeas 30803	Extend class notation to include the class of measures.
class measures
Definition	df-meas 30804*	Define a measure as a nonnegative countably additive function over a sigma-algebra onto (0[,]+∞). (Contributed by Thierry Arnoux, 10-Sep-2016.)
⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
Theorem	measbase 30805	The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
Theorem	measval 30806*	The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (measures‘𝑆) = {𝑚 ∣ (𝑚:𝑆⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
Theorem	ismeas 30807*	The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
Theorem	isrnmeas 30808*	The property of being a measure on an undefined base sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑀 ∈ ∪ ran measures → (dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
Theorem	dmmeas 30809	The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
Theorem	measbasedom 30810	The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
Theorem	measfrge0 30811	A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
Theorem	measfn 30812	A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)
Theorem	measvxrge0 30813	The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
Theorem	measvnul 30814	The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
Theorem	measge0 30815	A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (𝑀‘𝐴))
Theorem	measle0 30816	If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) = 0)
Theorem	measvun 30817*	The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))
Theorem	measxun2 30818	The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐴) = ((𝑀‘𝐵) +𝑒 (𝑀‘(𝐴 ∖ 𝐵))))
Theorem	measun 30819	The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵)))
Theorem	measvunilem 30820*	Lemma for measvuni 30822. (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑆 ∖ {∅}) ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵))
Theorem	measvunilem0 30821*	Lemma for measvuni 30822. (Contributed by Thierry Arnoux, 6-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ {∅} ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵))
Theorem	measvuni 30822*	The measure of a countable disjoint union is the sum of the measures. This theorem uses a collection rather than a set of subsets of 𝑆. (Contributed by Thierry Arnoux, 7-Mar-2017.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵))
Theorem	measssd 30823	A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	measunl 30824	A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵)))
Theorem	measiuns 30825*	The measure of the union of a collection of sets, expressed as the sum of a disjoint set. This is used as a lemma for both measiun 30826 and meascnbl 30827. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)))    &   ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
Theorem	measiun 30826*	A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵))
Theorem	meascnbl 30827*	A measure is continuous from below. Cf. volsup 23722. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    &   ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))    &   ⊢ (𝜑 → 𝐹:ℕ⟶𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))    ⇒   ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝𝑡‘𝐽)(𝑀‘∪ ran 𝐹))
Theorem	measinblem 30828*	Lemma for measinb 30829. (Contributed by Thierry Arnoux, 2-Jun-2017.)
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝐵 ∈ 𝒫 𝑆) ∧ (𝐵 ≼ ω ∧ Disj 𝑥 ∈ 𝐵 𝑥)) → (𝑀‘(∪ 𝐵 ∩ 𝐴)) = Σ*𝑥 ∈ 𝐵(𝑀‘(𝑥 ∩ 𝐴)))
Theorem	measinb 30829*	Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘𝑆))
Theorem	measres 30830	Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇))
Theorem	measinb2 30831*	Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ (𝑆 ∩ 𝒫 𝐴) ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘(𝑆 ∩ 𝒫 𝐴)))
Theorem	measdivcstOLD 30832*	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+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 𝐴)) ∈ (measures‘𝑆))
Theorem	measdivcst 30833	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+) → (𝑀∘𝑓/𝑐 /𝑒 𝐴) ∈ (measures‘𝑆))
20.3.16.7  The counting measure
Theorem	cntmeas 30834	The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (♯ ↾ 𝑆) ∈ (measures‘𝑆))
Theorem	pwcntmeas 30835	The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝑂 ∈ 𝑉 → (♯ ↾ 𝒫 𝑂) ∈ (measures‘𝒫 𝑂))
Theorem	cntnevol 30836	Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
⊢ (♯ ↾ 𝒫 𝑂) ≠ vol
20.3.16.8  The Lebesgue measure - misc additions
Theorem	voliune 30837	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 23671 and voliun 23720. (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 30838*	The Lebesgue measure function is countably additive. This theorem is to volfiniun 23713 what voliune 30837 is to voliun 23720. (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 30839	The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
⊢ vol ∈ (measures‘dom vol)
20.3.16.9  The Dirac delta measure
Syntax	cdde 30840	Extend class notation to include the Dirac delta measure.
class δ
Definition	df-dde 30841	Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
Theorem	ddeval1 30842	Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
Theorem	ddeval0 30843	Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)
Theorem	ddemeas 30844	The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ δ ∈ (measures‘𝒫 ℝ)
20.3.16.10  The 'almost everywhere' relation
Syntax	cae 30845	Extend class notation to include the 'almost everywhere' relation.
class a.e.
Syntax	cfae 30846	Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
Definition	df-ae 30847*	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 30848	'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
⊢ Rel a.e.
Theorem	brae 30849	'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 30850*	'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 30851*	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 30852*	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 30853*	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 30854*	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 30855	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 30856*	'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))
20.3.16.11  Measurable functions
Syntax	cmbfm 30857	Extend class notation with the measurable functions builder.
class MblFnM
Definition	df-mbfm 30858*	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 23785. (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 30859*	The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23794. (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 30860*	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 30861	A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	mbfmf 30862	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	isanmbfm 30863	The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmcnvima 30864	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	mbfmbfm 30865	A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
Theorem	mbfmcst 30866*	A constant function is measurable. Cf. mbfconst 23799. (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 30867	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 30868	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 30869*	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 30870	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 30871	The composition of two measurable functions is measurable. ( cf. cnmpt11 21837) (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 30872*	The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21839). (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 30873	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 30874	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 30875	All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑𝑚 𝑂))
20.3.16.12  Borel Algebra on ` ( RR X. RR ) `
Theorem	br2base 30876*	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 30877	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 30878*	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 30879*	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 30880*	The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23766. (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 30881*	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 30882*	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 30883*	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 30884*	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 30885*	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 30886*	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 30887*	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 30888*	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 30889*	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 30890*	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 30887. (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 30891*	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 30892*	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 30893*	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 30894*	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 30895*	First direction for sxbrsiga 30897. (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 30896	First direction for sxbrsiga 30897, 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 30897	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))
20.3.16.13  Caratheodory's extension theorem In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure 𝑅. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 30904) - it is monotone (omsmon 30905) - it is countably sub-additive (omssubadd 30907) See Definition 1.11.1 of [Bogachev] p. 41.
Syntax	coms 30898	Class declaration for the outer measure construction function.
class toOMeas
Definition	df-oms 30899*	Define a function constructing an outer measure. See omsval 30900 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 30900*	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[,]+∞), < )))