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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremispisys2 30201* 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 30202* 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 30203* 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 30204* 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 30205* 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 30206* 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 30207* 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 30208* 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 30209* Lemma for sigapildsys 30210. (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 30210* 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 30211* Lemma for ldgenpisys 30214. (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 30212* Lemma for ldgenpisys 30214. (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 30213* Lemma for ldgenpisys 30214. (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 30214* 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 30215* 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 30216* 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 30217* 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 30218* 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 30219* 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 30220* 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 30221* 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 30222* 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 30223* 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 30224* 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 30225* 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 30226* 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 30227* 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 30228* 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)

20.3.16.4  The Borel algebra on the real numbers

Syntaxcbrsiga 30229 The Borel Algebra on real numbers, usually a gothic B
class 𝔅

Definitiondf-brsiga 30230 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 30231 The Borel Algebra on real numbers is a Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 ∈ (sigaGen “ Top)

Theorembrsigarn 30232 The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 ∈ (sigAlgebra‘ℝ)

Theorembrsigasspwrn 30233 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
𝔅 ⊆ 𝒫 ℝ

Theoremunibrsiga 30234 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝔅 = ℝ

Theoremcldssbrsiga 30235 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
(𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))

20.3.16.5  Product Sigma-Algebra

Syntaxcsx 30236 Extend class notation with the product sigma-algebra operation.
class ×s

Definitiondf-sx 30237* Define the product sigma-algebra operation, analogous to df-tx 21359. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))

Theoremsxval 30238* 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 30239 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 30240 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 30241 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 30242 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))

20.3.16.6  Measures

Syntaxcmeas 30243 Extend class notation to include the class of measures.
class measures

Definitiondf-meas 30244* 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 30245 The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)

Theoremmeasval 30246* 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 30247* 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 30248* 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 30249 The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)
(𝑀 ran measures → dom 𝑀 ran sigAlgebra)

Theoremmeasbasedom 30250 The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))

Theoremmeasfrge0 30251 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 30252 A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
(𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)

Theoremmeasvxrge0 30253 The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑀𝐴) ∈ (0[,]+∞))

Theoremmeasvnul 30254 The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
(𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)

Theoremmeasge0 30255 A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → 0 ≤ (𝑀𝐴))

Theoremmeasle0 30256 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 30257* 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 30258 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 30259 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 30260* Lemma for measvuni 30262. (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 30261* Lemma for measvuni 30262. (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 30262* 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 30263 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 30264 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 30265* 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 30266 and meascnbl 30267. (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 30266* 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 30267* A measure is continuous from below. Cf. volsup 23318. (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 30268* Lemma for measinb 30269. (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)))

Theoremmeasinb 30269* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑥𝑆 ↦ (𝑀‘(𝑥𝐴))) ∈ (measures‘𝑆))

Theoremmeasres 30270 Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇) ∈ (measures‘𝑇))

Theoremmeasinb2 30271* Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆) → (𝑥 ∈ (𝑆 ∩ 𝒫 𝐴) ↦ (𝑀‘(𝑥𝐴))) ∈ (measures‘(𝑆 ∩ 𝒫 𝐴)))

TheoremmeasdivcstOLD 30272* 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+) → (𝑥𝑆 ↦ ((𝑀𝑥) /𝑒 𝐴)) ∈ (measures‘𝑆))

Theoremmeasdivcst 30273 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+) → (𝑀𝑓/𝑐 /𝑒 𝐴) ∈ (measures‘𝑆))

20.3.16.7  The counting measure

Theoremcntmeas 30274 The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑆 ran sigAlgebra → (# ↾ 𝑆) ∈ (measures‘𝑆))

Theorempwcntmeas 30275 The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝑂𝑉 → (# ↾ 𝒫 𝑂) ∈ (measures‘𝒫 𝑂))

Theoremcntnevol 30276 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

Theoremvoliune 30277 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 23267 and voliun 23316. (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))

Theoremvolfiniune 30278* The Lebesgue measure function is countably additive. This theorem is to volfiniun 23309 what voliune 30277 is to voliun 23316. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((𝐴 ∈ Fin ∧ ∀𝑛𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛𝐴 𝐵) → (vol‘ 𝑛𝐴 𝐵) = Σ*𝑛𝐴(vol‘𝐵))

Theoremvolmeas 30279 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

Syntaxcdde 30280 Extend class notation to include the Dirac delta measure.
class δ

Definitiondf-dde 30281 Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))

Theoremddeval1 30282 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)

Theoremddeval0 30283 Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)

Theoremddemeas 30284 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

Syntaxcae 30285 Extend class notation to include the 'almost everywhere' relation.
class a.e.

Syntaxcfae 30286 Extend class notation to include the 'almost everywhere' builder.
class ~ a.e.

Definitiondf-ae 30287* 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 𝑚𝑎)) = 0}

Theoremrelae 30288 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Rel a.e.

Theorembrae 30289 '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 𝑀𝐴)) = 0))

Theorembraew 30290* 'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.)
dom 𝑀 = 𝑂       (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))

Theoremtruae 30291* A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
dom 𝑀 = 𝑂    &   (𝜑𝑀 ran measures)    &   (𝜑𝜓)       (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)

Theoremaean 30292* 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 ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ ({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀)))

Definitiondf-fae 30293* 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 𝑟𝑚 dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟𝑚 dom 𝑚)) ∧ {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚)})

Theoremfaeval 30294* 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 𝑅𝑚 dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅𝑚 dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)})

Theoremrelfae 30295 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))

Theorembrfae 30296* '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    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ (𝐷𝑚 dom 𝑀))    &   (𝜑𝐺 ∈ (𝐷𝑚 dom 𝑀))       (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))

20.3.16.11  Measurable functions

Syntaxcmbfm 30297 Extend class notation with the measurable functions builder.
class MblFnM

Definitiondf-mbfm 30298* 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 to , see df-mbf 23382. (Contributed by Thierry Arnoux, 23-Jan-2017.)

MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})

Theoremismbfm 30299* The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23391. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))

Theoremelunirnmbfm 30300* The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))

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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 42301-42322
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