P. 349 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sigaclcu3 34801*	A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑁 𝐴 ∈ 𝑆)
Theorem	issgon 34802	Property of being a sigma-algebra with a given base set, noting that the base set of a sigma-algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
⊢ (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑂 = ∪ 𝑆))
Theorem	sgon 34803	A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
Theorem	elsigass 34804	An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
Theorem	elrnsiga 34805	Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)
Theorem	isrnsigau 34806*	The property of being a sigma-algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
Theorem	unielsiga 34807	A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
Theorem	dmvlsiga 34808	Lebesgue-measurable subsets of ℝ form a sigma-algebra. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
⊢ dom vol ∈ (sigAlgebra‘ℝ)
Theorem	pwsiga 34809	Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Theorem	prsiga 34810	The smallest possible sigma-algebra containing 𝑂. (Contributed by Thierry Arnoux, 13-Sep-2016.)
⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
Theorem	sigaclci 34811	A sigma-algebra is closed under countable intersections. Deduction version. The proof uses abrexct 23779 rather than abrexdom2jm 33152, and so does not require ax-ac 10541. (Contributed by Thierry Arnoux, 19-Sep-2016.) (Revised by Vincent Gonzalez, 19-Aug-2026.)
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)
Theorem	difunielsiga 34812	A sigma-algebra is closed under complement relative to its base set. This is immediate from the definition, see issiga 34791, but the library states it nowhere in this form. (Contributed by Vincent Gonzalez, 17-Aug-2026.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)
Theorem	unelsiga 34813	A sigma-algebra is closed under pairwise unions. (Contributed by Thierry Arnoux, 13-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	difelsiga 34814	A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34812 and unelsiga 34813 rather than countable intersection, and so does not use ax-ac 10541. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Proof shortened by Vincent Gonzalez, 17-Aug-2026.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
Theorem	inelsiga 34815	A sigma-algebra is closed under pairwise intersections. (Contributed by Thierry Arnoux, 13-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	sigainb 34816	Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Theorem	insiga 34817	The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))
21.4.18.2  Generated sigma-Algebra
Syntax	csigagen 34818	Extend class notation to include the sigma-algebra generator.
class sigaGen
Definition	df-sigagen 34819*	Define the sigma-algebra generated by a given collection of sets as the intersection of all sigma-algebra containing that set. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
Theorem	sigagenval 34820*	Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
Theorem	sigagensiga 34821	A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))
Theorem	sgsiga 34822	A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
Theorem	unisg 34823	The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴)
Theorem	dmsigagen 34824	A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ dom sigaGen = V
Theorem	sssigagen 34825	A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
Theorem	sssigagen2 34826	A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
Theorem	elsigagen 34827	Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴))
Theorem	elsigagen2 34828	Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
Theorem	sigagenss 34829	The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here 𝐴. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
Theorem	sigagenss2 34830	Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Theorem	sigagenid 34831	The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
21.4.18.3  lambda and pi-Systems, Rings of Sets Because they are not widely used outside of measure theory, we do not introduce specific definitions for lambda- and pi-systems. Instead, we define 𝑃 and 𝐿 respectively as the classes of pi- and lambda-systems in 𝑂 throughout this section.
Theorem	ispisys 34832*	The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Theorem	ispisys2 34833*	The property of being a pi-system, expanded version. Pi-systems are closed under finite intersections. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
Theorem	inelpisys 34834*	Pi-systems are closed under pairwise intersections. (Contributed by Thierry Arnoux, 6-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ ((𝑆 ∈ 𝑃 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	sigapisys 34835*	All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (sigAlgebra‘𝑂) ⊆ 𝑃
Theorem	isldsys 34836*	The property of being a lambda-system or Dynkin system. Lambda-systems contain the empty set, are closed under complement, and closed under countable disjoint union. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))))
Theorem	pwldsys 34837*	The power set of the universe set 𝑂 is always a lambda-system. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)
Theorem	unelldsys 34838*	Lambda-systems are closed under disjoint set unions. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	sigaldsys 34839*	All sigma-algebras are lambda-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿
Theorem	ldsysgenld 34840*	The intersection of all lambda-systems containing a given collection of sets 𝐴, which is called the lambda-system generated by 𝐴, is itself also a lambda-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝐿)
Theorem	sigapildsyslem 34841*	Lemma for sigapildsys 34842. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝜑 → 𝑡 ∈ (𝑃 ∩ 𝐿))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑡)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐵 ∈ 𝑡)    ⇒   ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑁 𝐵) ∈ 𝑡)
Theorem	sigapildsys 34842*	Sigma-algebra are exactly classes which are both lambda and pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (sigAlgebra‘𝑂) = (𝑃 ∩ 𝐿)
Theorem	ldgenpisyslem1 34843*	Lemma for ldgenpisys 34846. (Contributed by Thierry Arnoux, 29-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐸)    ⇒   ⊢ (𝜑 → {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸} ∈ 𝐿)
Theorem	ldgenpisyslem2 34844*	Lemma for ldgenpisys 34846. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑇 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})    ⇒   ⊢ (𝜑 → 𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
Theorem	ldgenpisyslem3 34845*	Lemma for ldgenpisys 34846. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    ⇒   ⊢ (𝜑 → 𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
Theorem	ldgenpisys 34846*	The lambda system 𝐸 generated by a pi-system 𝑇 is also a pi-system. (Contributed by Thierry Arnoux, 18-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐸 ∈ 𝑃)
Theorem	dynkin 34847*	Dynkin's lambda-pi theorem: if a lambda-system contains a pi-system, it also contains the sigma-algebra generated by that pi-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → ∩ {𝑢 ∈ (sigAlgebra‘𝑂) ∣ 𝑇 ⊆ 𝑢} ⊆ 𝑆)
Theorem	isros 34848*	The property of being a rings of sets, i.e. containing the empty set, and closed under finite union and set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
Theorem	rossspw 34849*	A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
Theorem	0elros 34850*	A ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
Theorem	unelros 34851*	A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	difelros 34852*	A ring of sets is closed under set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
Theorem	inelros 34853*	A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	fiunelros 34854*	A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝑄)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)
Theorem	issros 34855*	The property of being a semirings of sets, i.e., collections of sets containing the empty set, closed under finite intersection, and where complements can be written as finite disjoint unions. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))))
Theorem	srossspw 34856*	A semiring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑁 → 𝑆 ⊆ 𝒫 𝑂)
Theorem	0elsros 34857*	A semiring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑁 → ∅ ∈ 𝑆)
Theorem	inelsros 34858*	A semiring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	diffiunisros 34859*	In semiring of sets, complements can be written as finite disjoint unions. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧))
Theorem	rossros 34860*	Rings of sets are semirings of sets. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)
21.4.18.4  The Borel algebra on the real numbers
Syntax	cbrsiga 34861	The Borel Algebra on real numbers, usually a gothic B
class 𝔅ℝ
Definition	df-brsiga 34862	A Borel Algebra is defined as a sigma-algebra generated by a topology. 'The' Borel sigma-algebra here refers to the sigma-algebra generated by the topology of open intervals on real numbers. The Borel algebra of a given topology 𝐽 is the sigma-algebra generated by 𝐽, (sigaGen‘𝐽), so there is no need to introduce a special constant function for Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,)))
Theorem	brsiga 34863	The Borel Algebra on real numbers is a Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ 𝔅ℝ ∈ (sigaGen “ Top)
Theorem	brsigarn 34864	The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
Theorem	brsigasspwrn 34865	The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
⊢ 𝔅ℝ ⊆ 𝒫 ℝ
Theorem	unibrsiga 34866	The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ ∪ 𝔅ℝ = ℝ
Theorem	cldssbrsiga 34867	A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
21.4.18.5  Product Sigma-Algebra
Syntax	csx 34868	Extend class notation with the product sigma-algebra operation.
class ×s
Definition	df-sx 34869*	Define the product sigma-algebra operation, analogous to df-tx 23887. (Contributed by Thierry Arnoux, 1-Jun-2017.)
⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
Theorem	sxval 34870*	Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
⊢ 𝐴 = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Theorem	sxsiga 34871	A product sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
Theorem	sxsigon 34872	A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘(∪ 𝑆 × ∪ 𝑇)))
Theorem	sxuni 34873	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 34874	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))
21.4.18.6  Measures
Syntax	cmeas 34875	Extend class notation to include the class of measures.
class measures
Definition	df-meas 34876*	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 34877	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 34878*	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 34879*	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 34880*	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 34881	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 34882	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 34883	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 34884	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 34885	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 34886	The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
Theorem	measge0 34887	A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (𝑀‘𝐴))
Theorem	measle0 34888	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 34889*	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 34890	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 34891	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 34892*	Lemma for measvuni 34894. (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 34893*	Lemma for measvuni 34894. (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 34894*	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 34895	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 34896	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 34897*	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 34898 and meascnbl 34899. (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 34898*	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 34899*	A measure is continuous from below. Cf. volsup 25883. (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 34900*	Lemma for measinb 34901. (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)))