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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsigagensiga 32501 A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∈ 𝑉 β†’ (sigaGenβ€˜π΄) ∈ (sigAlgebraβ€˜βˆͺ 𝐴))
 
Theoremsgsiga 32502 A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (sigaGenβ€˜π΄) ∈ βˆͺ ran sigAlgebra)
 
Theoremunisg 32503 The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on βˆͺ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∈ 𝑉 β†’ βˆͺ (sigaGenβ€˜π΄) = βˆͺ 𝐴)
 
Theoremdmsigagen 32504 A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
dom sigaGen = V
 
Theoremsssigagen 32505 A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝐴 ∈ 𝑉 β†’ 𝐴 βŠ† (sigaGenβ€˜π΄))
 
Theoremsssigagen2 32506 A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
((𝐴 ∈ 𝑉 ∧ 𝐡 βŠ† 𝐴) β†’ 𝐡 βŠ† (sigaGenβ€˜π΄))
 
Theoremelsigagen 32507 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 ∧ 𝐡 ∈ 𝐴) β†’ 𝐡 ∈ (sigaGenβ€˜π΄))
 
Theoremelsigagen2 32508 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 ∧ 𝐡 βŠ† 𝐴 ∧ 𝐡 β‰Ό Ο‰) β†’ βˆͺ 𝐡 ∈ (sigaGenβ€˜π΄))
 
Theoremsigagenss 32509 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 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ (sigaGenβ€˜π΄) βŠ† 𝑆)
 
Theoremsigagenss2 32510 Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
((βˆͺ 𝐴 = βˆͺ 𝐡 ∧ 𝐴 βŠ† (sigaGenβ€˜π΅) ∧ 𝐡 ∈ 𝑉) β†’ (sigaGenβ€˜π΄) βŠ† (sigaGenβ€˜π΅))
 
Theoremsigagenid 32511 The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝑆 ∈ βˆͺ ran sigAlgebra β†’ (sigaGenβ€˜π‘†) = 𝑆)
 
21.3.17.3  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.

 
Theoremispisys 32512* The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    β‡’   (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fiβ€˜π‘†) βŠ† 𝑆))
 
Theoremispisys2 32513* 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 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    β‡’   (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ ((𝒫 𝑆 ∩ Fin) βˆ– {βˆ…})∩ π‘₯ ∈ 𝑆))
 
Theoreminelpisys 32514* Pi-systems are closed under pairwise intersections. (Contributed by Thierry Arnoux, 6-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    β‡’   ((𝑆 ∈ 𝑃 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 ∩ 𝐡) ∈ 𝑆)
 
Theoremsigapisys 32515* All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    β‡’   (sigAlgebraβ€˜π‘‚) βŠ† 𝑃
 
Theoremisldsys 32516* 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 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    β‡’   (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (βˆ… ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 (𝑂 βˆ– π‘₯) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑆))))
 
Theorempwldsys 32517* 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 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    β‡’   (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ 𝐿)
 
Theoremunelldsys 32518* Lambda-systems are closed under disjoint set unions. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑆 ∈ 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    β‡’   (πœ‘ β†’ (𝐴 βˆͺ 𝐡) ∈ 𝑆)
 
Theoremsigaldsys 32519* All sigma-algebras are lambda-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    β‡’   (sigAlgebraβ€˜π‘‚) βŠ† 𝐿
 
Theoremldsysgenld 32520* 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 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 βŠ† 𝒫 𝑂)    β‡’   (πœ‘ β†’ ∩ {𝑑 ∈ 𝐿 ∣ 𝐴 βŠ† 𝑑} ∈ 𝐿)
 
Theoremsigapildsyslem 32521* Lemma for sigapildsys 32522. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   β„²π‘›πœ‘    &   (πœ‘ β†’ 𝑑 ∈ (𝑃 ∩ 𝐿))    &   (πœ‘ β†’ 𝐴 ∈ 𝑑)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑛 ∈ 𝑁) β†’ 𝐡 ∈ 𝑑)    β‡’   (πœ‘ β†’ (𝐴 βˆ– βˆͺ 𝑛 ∈ 𝑁 𝐡) ∈ 𝑑)
 
Theoremsigapildsys 32522* Sigma-algebra are exactly classes which are both lambda and pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    β‡’   (sigAlgebraβ€˜π‘‚) = (𝑃 ∩ 𝐿)
 
Theoremldgenpisyslem1 32523* Lemma for ldgenpisys 32526. (Contributed by Thierry Arnoux, 29-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   πΈ = ∩ {𝑑 ∈ 𝐿 ∣ 𝑇 βŠ† 𝑑}    &   (πœ‘ β†’ 𝑇 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝐸)    β‡’   (πœ‘ β†’ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸} ∈ 𝐿)
 
Theoremldgenpisyslem2 32524* Lemma for ldgenpisys 32526. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   πΈ = ∩ {𝑑 ∈ 𝐿 ∣ 𝑇 βŠ† 𝑑}    &   (πœ‘ β†’ 𝑇 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝐸)    &   (πœ‘ β†’ 𝑇 βŠ† {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})    β‡’   (πœ‘ β†’ 𝐸 βŠ† {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
 
Theoremldgenpisyslem3 32525* Lemma for ldgenpisys 32526. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   πΈ = ∩ {𝑑 ∈ 𝐿 ∣ 𝑇 βŠ† 𝑑}    &   (πœ‘ β†’ 𝑇 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑇)    β‡’   (πœ‘ β†’ 𝐸 βŠ† {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
 
Theoremldgenpisys 32526* 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 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   πΈ = ∩ {𝑑 ∈ 𝐿 ∣ 𝑇 βŠ† 𝑑}    &   (πœ‘ β†’ 𝑇 ∈ 𝑃)    β‡’   (πœ‘ β†’ 𝐸 ∈ 𝑃)
 
Theoremdynkin 32527* 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 ∣ (fiβ€˜π‘ ) βŠ† 𝑠}    &   πΏ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 (𝑂 βˆ– π‘₯) ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ βˆͺ π‘₯ ∈ 𝑠))}    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ 𝐿)    &   (πœ‘ β†’ 𝑇 ∈ 𝑃)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑆)    β‡’   (πœ‘ β†’ ∩ {𝑒 ∈ (sigAlgebraβ€˜π‘‚) ∣ 𝑇 βŠ† 𝑒} βŠ† 𝑆)
 
Theoremisros 32528* 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 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ βˆ… ∈ 𝑆 ∧ βˆ€π‘’ ∈ 𝑆 βˆ€π‘£ ∈ 𝑆 ((𝑒 βˆͺ 𝑣) ∈ 𝑆 ∧ (𝑒 βˆ– 𝑣) ∈ 𝑆)))
 
Theoremrossspw 32529* A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   (𝑆 ∈ 𝑄 β†’ 𝑆 βŠ† 𝒫 𝑂)
 
Theorem0elros 32530* A ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   (𝑆 ∈ 𝑄 β†’ βˆ… ∈ 𝑆)
 
Theoremunelros 32531* A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 βˆͺ 𝐡) ∈ 𝑆)
 
Theoremdifelros 32532* A ring of sets is closed under set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 βˆ– 𝐡) ∈ 𝑆)
 
Theoreminelros 32533* A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    β‡’   ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 ∩ 𝐡) ∈ 𝑆)
 
Theoremfiunelros 32534* A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    &   (πœ‘ β†’ 𝑆 ∈ 𝑄)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   ((πœ‘ ∧ π‘˜ ∈ (1..^𝑁)) β†’ 𝐡 ∈ 𝑆)    β‡’   (πœ‘ β†’ βˆͺ π‘˜ ∈ (1..^𝑁)𝐡 ∈ 𝑆)
 
Theoremissros 32535* 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 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   (𝑆 ∈ 𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ βˆ… ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 ((π‘₯ ∩ 𝑦) ∈ 𝑆 ∧ βˆƒπ‘§ ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧))))
 
Theoremsrossspw 32536* A semiring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   (𝑆 ∈ 𝑁 β†’ 𝑆 βŠ† 𝒫 𝑂)
 
Theorem0elsros 32537* A semiring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   (𝑆 ∈ 𝑁 β†’ βˆ… ∈ 𝑆)
 
Theoreminelsros 32538* A semiring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 ∩ 𝐡) ∈ 𝑆)
 
Theoremdiffiunisros 32539* In semiring of sets, complements can be written as finite disjoint unions. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ βˆƒπ‘§ ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (𝐴 βˆ– 𝐡) = βˆͺ 𝑧))
 
Theoremrossros 32540* Rings of sets are semirings of sets. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ βˆͺ 𝑦) ∈ 𝑠 ∧ (π‘₯ βˆ– 𝑦) ∈ 𝑠))}    &   π‘ = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (βˆ… ∈ 𝑠 ∧ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑠 ((π‘₯ ∩ 𝑦) ∈ 𝑠 ∧ βˆƒπ‘§ ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑑 ∈ 𝑧 𝑑 ∧ (π‘₯ βˆ– 𝑦) = βˆͺ 𝑧)))}    β‡’   (𝑆 ∈ 𝑄 β†’ 𝑆 ∈ 𝑁)
 
21.3.17.4  The Borel algebra on the real numbers
 
Syntaxcbrsiga 32541 The Borel Algebra on real numbers, usually a gothic B
class 𝔅ℝ
 
Definitiondf-brsiga 32542 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, (sigaGenβ€˜π½), so there is no need to introduce a special constant function for Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅ℝ = (sigaGenβ€˜(topGenβ€˜ran (,)))
 
Theorembrsiga 32543 The Borel Algebra on real numbers is a Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅ℝ ∈ (sigaGen β€œ Top)
 
Theorembrsigarn 32544 The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅ℝ ∈ (sigAlgebraβ€˜β„)
 
Theorembrsigasspwrn 32545 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
𝔅ℝ βŠ† 𝒫 ℝ
 
Theoremunibrsiga 32546 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
βˆͺ 𝔅ℝ = ℝ
 
Theoremcldssbrsiga 32547 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
(𝐽 ∈ Top β†’ (Clsdβ€˜π½) βŠ† (sigaGenβ€˜π½))
 
21.3.17.5  Product Sigma-Algebra
 
Syntaxcsx 32548 Extend class notation with the product sigma-algebra operation.
class Γ—s
 
Definitiondf-sx 32549* Define the product sigma-algebra operation, analogous to df-tx 22835. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Γ—s = (𝑠 ∈ V, 𝑑 ∈ V ↦ (sigaGenβ€˜ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))))
 
Theoremsxval 32550* Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
𝐴 = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))    β‡’   ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ π‘Š) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜π΄))
 
Theoremsxsiga 32551 A product sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra)
 
Theoremsxsigon 32552 A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜(βˆͺ 𝑆 Γ— βˆͺ 𝑇)))
 
Theoremsxuni 32553 The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (βˆͺ 𝑆 Γ— βˆͺ 𝑇) = βˆͺ (𝑆 Γ—s 𝑇))
 
Theoremelsx 32554 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 ∈ π‘Š) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑇)) β†’ (𝐴 Γ— 𝐡) ∈ (𝑆 Γ—s 𝑇))
 
21.3.17.6  Measures
 
Syntaxcmeas 32555 Extend class notation to include the class of measures.
class measures
 
Definitiondf-meas 32556* Define a measure as a nonnegative countably additive function over a sigma-algebra onto (0[,]+∞). (Contributed by Thierry Arnoux, 10-Sep-2016.)
measures = (𝑠 ∈ βˆͺ ran sigAlgebra ↦ {π‘š ∣ (π‘š:π‘ βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
 
Theoremmeasbase 32557 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
 
Theoremmeasval 32558* The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(𝑆 ∈ βˆͺ ran sigAlgebra β†’ (measuresβ€˜π‘†) = {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
 
Theoremismeas 32559* The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
(𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦)))))
 
Theoremisrnmeas 32560* The property of being a measure on an undefined base sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ∈ βˆͺ ran measures β†’ (dom 𝑀 ∈ βˆͺ ran sigAlgebra ∧ (𝑀:dom π‘€βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 dom 𝑀((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦)))))
 
Theoremdmmeas 32561 The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
(𝑀 ∈ βˆͺ ran measures β†’ dom 𝑀 ∈ βˆͺ ran sigAlgebra)
 
Theoremmeasbasedom 32562 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ∈ βˆͺ ran measures ↔ 𝑀 ∈ (measuresβ€˜dom 𝑀))
 
Theoremmeasfrge0 32563 A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
(𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
 
Theoremmeasfn 32564 A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
(𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑀 Fn 𝑆)
 
Theoremmeasvxrge0 32565 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆) β†’ (π‘€β€˜π΄) ∈ (0[,]+∞))
 
Theoremmeasvnul 32566 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
(𝑀 ∈ (measuresβ€˜π‘†) β†’ (π‘€β€˜βˆ…) = 0)
 
Theoremmeasge0 32567 A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆) β†’ 0 ≀ (π‘€β€˜π΄))
 
Theoremmeasle0 32568 If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆 ∧ (π‘€β€˜π΄) ≀ 0) β†’ (π‘€β€˜π΄) = 0)
 
Theoremmeasvun 32569* The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))
 
Theoremmeasxun2 32570 The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) ∧ 𝐡 βŠ† 𝐴) β†’ (π‘€β€˜π΄) = ((π‘€β€˜π΅) +𝑒 (π‘€β€˜(𝐴 βˆ– 𝐡))))
 
Theoremmeasun 32571 The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (π‘€β€˜(𝐴 βˆͺ 𝐡)) = ((π‘€β€˜π΄) +𝑒 (π‘€β€˜π΅)))
 
Theoremmeasvunilem 32572* Lemma for measvuni 32574. (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.)
β„²π‘₯𝐴    β‡’   ((𝑀 ∈ (measuresβ€˜π‘†) ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ (𝑆 βˆ– {βˆ…}) ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 𝐡)) β†’ (π‘€β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π΅))
 
Theoremmeasvunilem0 32573* Lemma for measvuni 32574. (Contributed by Thierry Arnoux, 6-Mar-2017.)
β„²π‘₯𝐴    β‡’   ((𝑀 ∈ (measuresβ€˜π‘†) ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ {βˆ…} ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 𝐡)) β†’ (π‘€β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π΅))
 
Theoremmeasvuni 32574* 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 ∈ (measuresβ€˜π‘†) ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 𝐡)) β†’ (π‘€β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π΅))
 
Theoremmeasssd 32575 A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(πœ‘ β†’ 𝑀 ∈ (measuresβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (π‘€β€˜π΄) ≀ (π‘€β€˜π΅))
 
Theoremmeasunl 32576 A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
(πœ‘ β†’ 𝑀 ∈ (measuresβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜(𝐴 βˆͺ 𝐡)) ≀ ((π‘€β€˜π΄) +𝑒 (π‘€β€˜π΅)))
 
Theoremmeasiuns 32577* 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 32578 and meascnbl 32579. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
Ⅎ𝑛𝐡    &   (𝑛 = π‘˜ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ (𝑁 = β„• ∨ 𝑁 = (1..^𝐼)))    &   (πœ‘ β†’ 𝑀 ∈ (measuresβ€˜π‘†))    &   ((πœ‘ ∧ 𝑛 ∈ 𝑁) β†’ 𝐴 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜βˆͺ 𝑛 ∈ 𝑁 𝐴) = Ξ£*𝑛 ∈ 𝑁(π‘€β€˜(𝐴 βˆ– βˆͺ π‘˜ ∈ (1..^𝑛)𝐡)))
 
Theoremmeasiun 32578* A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
(πœ‘ β†’ 𝑀 ∈ (measuresβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝑛 ∈ β„• 𝐡)    β‡’   (πœ‘ β†’ (π‘€β€˜π΄) ≀ Ξ£*𝑛 ∈ β„•(π‘€β€˜π΅))
 
Theoremmeascnbl 32579* A measure is continuous from below. Cf. volsup 24842. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
𝐽 = (TopOpenβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))    &   (πœ‘ β†’ 𝑀 ∈ (measuresβ€˜π‘†))    &   (πœ‘ β†’ 𝐹:β„•βŸΆπ‘†)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) βŠ† (πΉβ€˜(𝑛 + 1)))    β‡’   (πœ‘ β†’ (𝑀 ∘ 𝐹)(β‡π‘‘β€˜π½)(π‘€β€˜βˆͺ ran 𝐹))
 
Theoremmeasinblem 32580* Lemma for measinb 32581. (Contributed by Thierry Arnoux, 2-Jun-2017.)
((((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆) ∧ 𝐡 ∈ 𝒫 𝑆) ∧ (𝐡 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐡 π‘₯)) β†’ (π‘€β€˜(βˆͺ 𝐡 ∩ 𝐴)) = Ξ£*π‘₯ ∈ 𝐡(π‘€β€˜(π‘₯ ∩ 𝐴)))
 
Theoremmeasinb 32581* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘€β€˜(π‘₯ ∩ 𝐴))) ∈ (measuresβ€˜π‘†))
 
Theoremmeasres 32582 Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (𝑀 β†Ύ 𝑇) ∈ (measuresβ€˜π‘‡))
 
Theoremmeasinb2 32583* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆) β†’ (π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴) ↦ (π‘€β€˜(π‘₯ ∩ 𝐴))) ∈ (measuresβ€˜(𝑆 ∩ 𝒫 𝐴)))
 
Theoremmeasdivcst 32584 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 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ ℝ+) β†’ (𝑀 ∘f/c /𝑒 𝐴) ∈ (measuresβ€˜π‘†))
 
TheoremmeasdivcstALTV 32585* Alternate version of measdivcst 32584. (Contributed by Thierry Arnoux, 25-Dec-2016.) (New usage is discouraged.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ ℝ+) β†’ (π‘₯ ∈ 𝑆 ↦ ((π‘€β€˜π‘₯) /𝑒 𝐴)) ∈ (measuresβ€˜π‘†))
 
21.3.17.7  The counting measure
 
Theoremcntmeas 32586 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑆 ∈ βˆͺ ran sigAlgebra β†’ (β™― β†Ύ 𝑆) ∈ (measuresβ€˜π‘†))
 
Theorempwcntmeas 32587 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝑂 ∈ 𝑉 β†’ (β™― β†Ύ 𝒫 𝑂) ∈ (measuresβ€˜π’« 𝑂))
 
Theoremcntnevol 32588 Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
(β™― β†Ύ 𝒫 𝑂) β‰  vol
 
21.3.17.8  The Lebesgue measure - misc additions
 
Theoremvoliune 32589 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 24791 and voliun 24840. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((βˆ€π‘› ∈ β„• 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ β„• 𝐴) β†’ (volβ€˜βˆͺ 𝑛 ∈ β„• 𝐴) = Ξ£*𝑛 ∈ β„•(volβ€˜π΄))
 
Theoremvolfiniune 32590* The Lebesgue measure function is countably additive. This theorem is to volfiniun 24833 what voliune 32589 is to voliun 24840. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((𝐴 ∈ Fin ∧ βˆ€π‘› ∈ 𝐴 𝐡 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐡) β†’ (volβ€˜βˆͺ 𝑛 ∈ 𝐴 𝐡) = Ξ£*𝑛 ∈ 𝐴(volβ€˜π΅))
 
Theoremvolmeas 32591 The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.)
vol ∈ (measuresβ€˜dom vol)
 
21.3.17.9  The Dirac delta measure
 
Syntaxcdde 32592 Extend class notation to include the Dirac delta measure.
class Ξ΄
 
Definitiondf-dde 32593 Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Ξ΄ = (π‘Ž ∈ 𝒫 ℝ ↦ if(0 ∈ π‘Ž, 1, 0))
 
Theoremddeval1 32594 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 βŠ† ℝ ∧ 0 ∈ 𝐴) β†’ (Ξ΄β€˜π΄) = 1)
 
Theoremddeval0 32595 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 βŠ† ℝ ∧ Β¬ 0 ∈ 𝐴) β†’ (Ξ΄β€˜π΄) = 0)
 
Theoremddemeas 32596 The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Ξ΄ ∈ (measuresβ€˜π’« ℝ)
 
21.3.17.10  The 'almost everywhere' relation
 
Syntaxcae 32597 Extend class notation to include the 'almost everywhere' relation.
class a.e.
 
Syntaxcfae 32598 Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.
 
Definitiondf-ae 32599* 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. = {βŸ¨π‘Ž, π‘šβŸ© ∣ (π‘šβ€˜(βˆͺ dom π‘š βˆ– π‘Ž)) = 0}
 
Theoremrelae 32600 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Rel a.e.
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