HomeHome Metamath Proof Explorer
Theorem List (p. 294 of 425)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26941)
  Hilbert Space Explorer  Hilbert Space Explorer
(26942-28466)
  Users' Mathboxes  Users' Mathboxes
(28467-42420)
 

Theorem List for Metamath Proof Explorer - 29301-29400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsigaclfu2 29301* A sigma-algebra is closed under finite union - indexing on (1..^𝑁). (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝑆 ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑁)𝐴𝑆) → 𝑘 ∈ (1..^𝑁)𝐴𝑆)
 
Theoremsigaclcu3 29302* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    &   ((𝜑𝑘𝑁) → 𝐴𝑆)       (𝜑 𝑘𝑁 𝐴𝑆)
 
Theoremissgon 29303 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))
 
Theoremsgon 29304 A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
(𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
 
Theoremelsigass 29305 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 𝑆)
 
Theoremelrnsiga 29306 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
(𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
 
Theoremisrnsigau 29307* 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 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
 
Theoremunielsiga 29308 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
(𝑆 ran sigAlgebra → 𝑆𝑆)
 
Theoremdmvlsiga 29309 Lebesgue-measurable subsets of form a sigma-algebra. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
dom vol ∈ (sigAlgebra‘ℝ)
 
Theorempwsiga 29310 Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
(𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
 
Theoremprsiga 29311 The smallest possible sigma-algebra containing 𝑂. (Contributed by Thierry Arnoux, 13-Sep-2016.)
(𝑂𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
 
Theoremsigaclci 29312 A sigma-algebra is closed under countable intersections. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
(((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)
 
Theoremdifelsiga 29313 A sigma-algebra is closed under class differences. (Contributed by Thierry Arnoux, 13-Sep-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremunelsiga 29314 A sigma-algebra is closed under pairwise unions. (Contributed by Thierry Arnoux, 13-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoreminelsiga 29315 A sigma-algebra is closed under pairwise intersections. (Contributed by Thierry Arnoux, 13-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremsigainb 29316 Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
 
Theoreminsiga 29317 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))
 
20.3.15.2  Generated sigma-Algebra
 
Syntaxcsigagen 29318 Extend class notation to include the sigma-algebra generator.
class sigaGen
 
Definitiondf-sigagen 29319* 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) ∣ 𝑥𝑠})
 
Theoremsigagenval 29320* Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
 
Theoremsigagensiga 29321 A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
 
Theoremsgsiga 29322 A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑𝐴𝑉)       (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)
 
Theoremunisg 29323 The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴𝑉 (sigaGen‘𝐴) = 𝐴)
 
Theoremdmsigagen 29324 A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
dom sigaGen = V
 
Theoremsssigagen 29325 A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
 
Theoremsssigagen2 29326 A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
 
Theoremelsigagen 29327 Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
((𝐴𝑉𝐵𝐴) → 𝐵 ∈ (sigaGen‘𝐴))
 
Theoremelsigagen2 29328 Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
 
Theoremsigagenss 29329 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) ∧ 𝐴𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
 
Theoremsigagenss2 29330 Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
(( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
 
Theoremsigagenid 29331 The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝑆 ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
 
20.3.15.3  lambda and pi-Systems, Rings of Sets

Because they are not widely used outside of measure theory, we don't introduce specific definitions for lambda- and pi-systems. Instead, we are defining 𝑃 and 𝐿 respectively as the classes of pi- and lambda-systems in 𝑂 throughout this section.

 
Theoremispisys 29332* The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}       (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
 
Theoremispisys2 29333* 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}       (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
 
Theoreminelpisys 29334* Pi-systems are closed under pairwise intersections. (Contributed by Thierry Arnoux, 6-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}       ((𝑆𝑃𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremsigapisys 29335* All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}       (sigAlgebra‘𝑂) ⊆ 𝑃
 
Theoremisldsys 29336* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}       (𝑆𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆))))
 
Theorempwldsys 29337* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}       (𝑂𝑉 → 𝒫 𝑂𝐿)
 
Theoremunelldsys 29338* Lambda-systems are closed under disjoint set unions. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑆𝐿)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝐴𝐵) ∈ 𝑆)
 
Theoremsigaldsys 29339* All sigma-algebras are lambda-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}       (sigAlgebra‘𝑂) ⊆ 𝐿
 
Theoremldsysgenld 29340* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   (𝜑𝐴 ⊆ 𝒫 𝑂)       (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
 
Theoremsigapildsyslem 29341* Lemma for sigapildsys 29342. (Contributed by Thierry Arnoux, 13-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   𝑛𝜑    &   (𝜑𝑡 ∈ (𝑃𝐿))    &   (𝜑𝐴𝑡)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑛𝑁) → 𝐵𝑡)       (𝜑 → (𝐴 𝑛𝑁 𝐵) ∈ 𝑡)
 
Theoremsigapildsys 29342* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}       (sigAlgebra‘𝑂) = (𝑃𝐿)
 
Theoremldgenpisyslem1 29343* Lemma for ldgenpisys 29346. (Contributed by Thierry Arnoux, 29-Jun-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   𝐸 = {𝑡𝐿𝑇𝑡}    &   (𝜑𝑇𝑃)    &   (𝜑𝐴𝐸)       (𝜑 → {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴𝑏) ∈ 𝐸} ∈ 𝐿)
 
Theoremldgenpisyslem2 29344* Lemma for ldgenpisys 29346. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   𝐸 = {𝑡𝐿𝑇𝑡}    &   (𝜑𝑇𝑃)    &   (𝜑𝐴𝐸)    &   (𝜑𝑇 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴𝑏) ∈ 𝐸})       (𝜑𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴𝑏) ∈ 𝐸})
 
Theoremldgenpisyslem3 29345* Lemma for ldgenpisys 29346. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   𝐸 = {𝑡𝐿𝑇𝑡}    &   (𝜑𝑇𝑃)    &   (𝜑𝐴𝑇)       (𝜑𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴𝑏) ∈ 𝐸})
 
Theoremldgenpisys 29346* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   𝐸 = {𝑡𝐿𝑇𝑡}    &   (𝜑𝑇𝑃)       (𝜑𝐸𝑃)
 
Theoremdynkin 29347* 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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}    &   (𝜑𝑂𝑉)    &   (𝜑𝑆𝐿)    &   (𝜑𝑇𝑃)    &   (𝜑𝑇𝑆)       (𝜑 {𝑢 ∈ (sigAlgebra‘𝑂) ∣ 𝑇𝑢} ⊆ 𝑆)
 
Theoremisros 29348* 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 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
 
Theoremrossspw 29349* A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
 
Theorem0elros 29350* A ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       (𝑆𝑄 → ∅ ∈ 𝑆)
 
Theoremunelros 29351* A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremdifelros 29352* A ring of sets is closed under set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoreminelros 29353* A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}       ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremfiunelros 29354* A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}    &   (𝜑𝑆𝑄)    &   (𝜑𝑁 ∈ ℕ)    &   ((𝜑𝑘 ∈ (1..^𝑁)) → 𝐵𝑆)       (𝜑 𝑘 ∈ (1..^𝑁)𝐵𝑆)
 
Theoremissros 29355* The property of being a semi-rings 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 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
 
Theoremsrossspw 29356* A semi-ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       (𝑆𝑁𝑆 ⊆ 𝒫 𝑂)
 
Theorem0elsros 29357* A semi-ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       (𝑆𝑁 → ∅ ∈ 𝑆)
 
Theoreminelsros 29358* A semi-ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       ((𝑆𝑁𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
 
Theoremdiffiunisros 29359* 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 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       ((𝑆𝑁𝐴𝑆𝐵𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧))
 
Theoremrossros 29360* Rings of sets are semi-rings of sets. (Contributed by Thierry Arnoux, 18-Jul-2020.)
𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}    &   𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}       (𝑆𝑄𝑆𝑁)
 
20.3.15.4  The Borel algebra on the real numbers
 
Syntaxcbrsiga 29361 The Borel Algebra on real numbers, usually a gothic B
class 𝔅
 
Definitiondf-brsiga 29362 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.)
𝔅 = (sigaGen‘(topGen‘ran (,)))
 
Theorembrsiga 29363 The Borel Algebra on real numbers is a Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 ∈ (sigaGen “ Top)
 
Theorembrsigarn 29364 The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 ∈ (sigAlgebra‘ℝ)
 
Theorembrsigasspwrn 29365 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
𝔅 ⊆ 𝒫 ℝ
 
Theoremunibrsiga 29366 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝔅 = ℝ
 
Theoremcldssbrsiga 29367 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
(𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
 
20.3.15.5  Product Sigma-Algebra
 
Syntaxcsx 29368 Extend class notation with the product sigma-algebra operation.
class ×s
 
Definitiondf-sx 29369* Define the product sigma-algebra operation, analogous to df-tx 21081. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
 
Theoremsxval 29370* Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))       ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
 
Theoremsxsiga 29371 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)
 
Theoremsxsigon 29372 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)))
 
Theoremsxuni 29373 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))
 
Theoremelsx 29374 The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
 
20.3.15.6  Measures
 
Syntaxcmeas 29375 Extend class notation to include the class of measures.
class measures
 
Definitiondf-meas 29376* 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 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
 
Theoremmeasbase 29377 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
 
Theoremmeasval 29378* 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 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
 
Theoremismeas 29379* 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 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
 
Theoremisrnmeas 29380* 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 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
 
Theoremdmmeas 29381 The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
(𝑀 ran measures → dom 𝑀 ran sigAlgebra)
 
Theoremmeasbasedom 29382 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
 
Theoremmeasfrge0 29383 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[,]+∞))
 
Theoremmeasfn 29384 A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
(𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)
 
Theoremmeasvxrge0 29385 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑀𝐴) ∈ (0[,]+∞))
 
Theoremmeasvnul 29386 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
(𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
 
Theoremmeasge0 29387 A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → 0 ≤ (𝑀𝐴))
 
Theoremmeasle0 29388 If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆 ∧ (𝑀𝐴) ≤ 0) → (𝑀𝐴) = 0)
 
Theoremmeasvun 29389* 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 𝑥𝐴 𝑥)) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))
 
Theoremmeasxun2 29390 The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((𝑀 ∈ (measures‘𝑆) ∧ (𝐴𝑆𝐵𝑆) ∧ 𝐵𝐴) → (𝑀𝐴) = ((𝑀𝐵) +𝑒 (𝑀‘(𝐴𝐵))))
 
Theoremmeasun 29391 The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((𝑀 ∈ (measures‘𝑆) ∧ (𝐴𝑆𝐵𝑆) ∧ (𝐴𝐵) = ∅) → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) +𝑒 (𝑀𝐵)))
 
Theoremmeasvunilem 29392* Lemma for measvuni 29394. (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.)
𝑥𝐴       ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥𝐴 𝐵 ∈ (𝑆 ∖ {∅}) ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝐵)) → (𝑀 𝑥𝐴 𝐵) = Σ*𝑥𝐴(𝑀𝐵))
 
Theoremmeasvunilem0 29393* Lemma for measvuni 29394. (Contributed by Thierry Arnoux, 6-Mar-2017.)
𝑥𝐴       ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥𝐴 𝐵 ∈ {∅} ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝐵)) → (𝑀 𝑥𝐴 𝐵) = Σ*𝑥𝐴(𝑀𝐵))
 
Theoremmeasvuni 29394* 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) ∧ ∀𝑥𝐴 𝐵𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝐵)) → (𝑀 𝑥𝐴 𝐵) = Σ*𝑥𝐴(𝑀𝐵))
 
Theoremmeasssd 29395 A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(𝜑𝑀 ∈ (measures‘𝑆))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
Theoremmeasunl 29396 A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)
(𝜑𝑀 ∈ (measures‘𝑆))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝑀‘(𝐴𝐵)) ≤ ((𝑀𝐴) +𝑒 (𝑀𝐵)))
 
Theoremmeasiuns 29397* 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 29398 and meascnbl 29399. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)))    &   (𝜑𝑀 ∈ (measures‘𝑆))    &   ((𝜑𝑛𝑁) → 𝐴𝑆)       (𝜑 → (𝑀 𝑛𝑁 𝐴) = Σ*𝑛𝑁(𝑀‘(𝐴 𝑘 ∈ (1..^𝑛)𝐵)))
 
Theoremmeasiun 29398* A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)
(𝜑𝑀 ∈ (measures‘𝑆))    &   (𝜑𝐴𝑆)    &   ((𝜑𝑛 ∈ ℕ) → 𝐵𝑆)    &   (𝜑𝐴 𝑛 ∈ ℕ 𝐵)       (𝜑 → (𝑀𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀𝐵))
 
Theoremmeascnbl 29399* A measure is continuous from below. Cf. volsup 23009. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   (𝜑𝑀 ∈ (measures‘𝑆))    &   (𝜑𝐹:ℕ⟶𝑆)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       (𝜑 → (𝑀𝐹)(⇝𝑡𝐽)(𝑀 ran 𝐹))
 
Theoremmeasinblem 29400* Lemma for measinb 29401. (Contributed by Thierry Arnoux, 2-Jun-2017.)
((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) ∧ 𝐵 ∈ 𝒫 𝑆) ∧ (𝐵 ≼ ω ∧ Disj 𝑥𝐵 𝑥)) → (𝑀‘( 𝐵𝐴)) = Σ*𝑥𝐵(𝑀‘(𝑥𝐴)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42420
  Copyright terms: Public domain < Previous  Next >