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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | meaiininclem 46501* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) & ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) & ⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
| Theorem | meaiininc 46502* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) & ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
| Theorem | meaiininc2 46503* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Proofs for most of the theorems in section 113 of [Fremlin1] | ||
| Syntax | come 46504 | Extend class notation with the class of outer measures. |
| class OutMeas | ||
| Definition | df-ome 46505* | Define the class of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))))} | ||
| Syntax | ccaragen 46506 | Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure. |
| class CaraGen | ||
| Definition | df-caragen 46507* | Define the sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) | ||
| Theorem | caragenval 46508* | The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) | ||
| Theorem | isome 46509* | Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))) | ||
| Theorem | caragenel 46510* | Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) | ||
| Theorem | omef 46511 | An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 ⇒ ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) | ||
| Theorem | ome0 46512 | The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) ⇒ ⊢ (𝜑 → (𝑂‘∅) = 0) | ||
| Theorem | omessle 46513 | The outer measure of a set is greater than or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐵 ⊆ 𝑋) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵)) | ||
| Theorem | omedm 46514 | The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂) | ||
| Theorem | caragensplit 46515 | If 𝐸 is in the set generated by the Caratheodory's method, then it splits any set 𝐴 in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set 𝐴. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) | ||
| Theorem | caragenelss 46516 | An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ 𝑋 = ∪ dom 𝑂 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | ||
| Theorem | carageneld 46517* | Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
| Theorem | omecl 46518 | The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) | ||
| Theorem | caragenss 46519 | The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) | ||
| Theorem | omeunile 46520 | The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) & ⊢ (𝜑 → 𝑌 ≼ ω) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))) | ||
| Theorem | caragen0 46521 | The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑆) | ||
| Theorem | omexrcl 46522 | The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) | ||
| Theorem | caragenunidm 46523 | The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑆) | ||
| Theorem | caragensspw 46524 | The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋) | ||
| Theorem | omessre 46525 | If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) | ||
| Theorem | caragenuni 46526 | The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂) | ||
| Theorem | caragenuncllem 46527 | The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴)) | ||
| Theorem | caragenuncl 46528 | The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) | ||
| Theorem | caragendifcl 46529 | The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | ||
| Theorem | caragenfiiuncl 46530* | The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
| Theorem | omeunle 46531 | The outer measure of the union of two sets is less than or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐵 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) | ||
| Theorem | omeiunle 46532* | The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑛𝐸 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))) | ||
| Theorem | omelesplit 46533 | The outer measure of a set 𝐴 is less than or equal to the extended addition of the outer measures of the decomposition induced on 𝐴 by any 𝐸. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) | ||
| Theorem | omeiunltfirp 46534* | If the outer measure of a countable union is not +∞, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋) & ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)) | ||
| Theorem | omeiunlempt 46535* | The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) | ||
| Theorem | carageniuncllem1 46536* | The outer measure of 𝐴 ∩ (𝐺‘𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹‘𝑚). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖))) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾)))) | ||
| Theorem | carageniuncllem2 46537* | The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖))) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌)) | ||
| Theorem | carageniuncl 46538* | The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝑆) | ||
| Theorem | caragenunicl 46539 | The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝑋 ≼ ω) ⇒ ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) | ||
| Theorem | caragensal 46540 | Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
| Theorem | caratheodorylem1 46541* | Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛)) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) | ||
| Theorem | caratheodorylem2 46542* | Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸:ℕ⟶𝑆) & ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛)) & ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))))) | ||
| Theorem | caratheodory 46543 | Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → (𝑂 ↾ 𝑆) ∈ Meas) | ||
| Theorem | 0ome 46544* | The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → 𝑂 ∈ OutMeas) | ||
| Theorem | isomenndlem 46545* | 𝑂 is sub-additive w.r.t. countable indexed union, implies that 𝑂 is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑂‘∅) = 0) & ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))))) & ⊢ (𝜑 → 𝐵 ⊆ ℕ) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑌) & ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))) | ||
| Theorem | isomennd 46546* | Sufficient condition to prove that 𝑂 is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑂‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥) → (𝑂‘𝑦) ≤ (𝑂‘𝑥)) & ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))))) ⇒ ⊢ (𝜑 → 𝑂 ∈ OutMeas) | ||
| Theorem | caragenel2d 46547* | Membership in the Caratheodory's construction. Similar to carageneld 46517, but here "less then or equal to" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
| Theorem | omege0 46548 | If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴)) | ||
| Theorem | omess0 46549 | If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) = 0) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑂‘𝐵) = 0) | ||
| Theorem | caragencmpl 46550 | A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐸 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐸) = 0) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
Proofs for most of the theorems in section 115 of [Fremlin1] | ||
| Syntax | covoln 46551 | Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers. |
| class voln* | ||
| Definition | df-ovoln 46552* | Define the outer measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑥) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )))) | ||
| Syntax | cvoln 46553 | Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers. |
| class voln | ||
| Definition | df-voln 46554 | Define the Lebesgue measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))) | ||
| Theorem | vonval 46555 | Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) | ||
| Theorem | ovnval 46556* | Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )))) | ||
| Theorem | elhoi 46557* | Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) | ||
| Theorem | icoresmbl 46558 | A closed-below, open-above real interval is measurable, when the bounds are real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol | ||
| Theorem | hoissre 46559* | The projection of a half-open interval onto a single dimension is a subset of ℝ. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) | ||
| Theorem | ovnval2 46560* | Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < ))) | ||
| Theorem | volicorecl 46561 | The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ) | ||
| Theorem | hoiprodcl 46562* | The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) | ||
| Theorem | hoicvr 46563* | 𝐼 is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ 〈-𝑗, 𝑗〉)) & ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (ℝ ↑m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)) | ||
| Theorem | hoissrrn 46564* | A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑m 𝑋)) | ||
| Theorem | ovn0val 46565 | The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m ∅)) ⇒ ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) | ||
| Theorem | ovnn0val 46566* | The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) | ||
| Theorem | ovnval2b 46567* | Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf((𝐿‘𝐴), ℝ*, < ))) | ||
| Theorem | volicorescl 46568 | The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) → (vol‘𝐴) ∈ ℝ) | ||
| Theorem | ovnprodcl 46569* | The product used in the definition of the outer Lebesgue measure in R^n is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐹:ℕ⟶((ℝ × ℝ) ↑m 𝑋)) & ⊢ (𝜑 → 𝐼 ∈ ℕ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐹‘𝐼))‘𝑘)) ∈ (0[,)+∞)) | ||
| Theorem | hoiprodcl2 46570* | The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) & ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ (𝜑 → (𝐿‘𝐼) ∈ (0[,)+∞)) | ||
| Theorem | hoicvrrex 46571* | Any subset of the multidimensional reals can be covered by a countable set of half-open intervals, see Definition 115A (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑m 𝑋)) ⇒ ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) | ||
| Theorem | ovnsupge0 46572* | The set used in the definition of the Lebesgue outer measure is a subset of the nonnegative extended reals. This is a substep for (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → 𝑀 ⊆ (0[,]+∞)) | ||
| Theorem | ovnlecvr 46573* | Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with 𝑋 the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) & ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))) | ||
| Theorem | ovnpnfelsup 46574* | +∞ is an element of the set used in the definition of the Lebesgue outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → +∞ ∈ 𝑀) | ||
| Theorem | ovnsslelem 46575* | The (multidimensional, nonzero-dimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} & ⊢ 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐵 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) | ||
| Theorem | ovnssle 46576 | The (multidimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵)) | ||
| Theorem | ovnlerp 46577* | The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) | ||
| Theorem | ovnf 46578 | The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞)) | ||
| Theorem | ovncvrrp 46579* | The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) & ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) & ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) ⇒ ⊢ (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸)) | ||
| Theorem | ovn0lem 46580* | For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (a)(ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))} & ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) & ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉)) ⇒ ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0) | ||
| Theorem | ovn0 46581 | For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘∅) = 0) | ||
| Theorem | ovncl 46582 | The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ (0[,]+∞)) | ||
| Theorem | ovn02 46583 | For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0) | ||
| Theorem | ovnxrcl 46584 | The Lebesgue outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ*) | ||
| Theorem | ovnsubaddlem1 46585* | The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}) & ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)}) & ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) & ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) & ⊢ (𝜑 → 𝐹:ℕ–1-1-onto→(ℕ × ℕ)) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) | ||
| Theorem | ovnsubaddlem2 46586* | (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}) & ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) & ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) & ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) | ||
| Theorem | ovnsubadd 46587* | (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋)) ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) | ||
| Theorem | ovnome 46588 | (voln*‘𝑋) is an outer measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set 𝑋. Proposition 115D (a) of [Fremlin1] p. 30 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) | ||
| Theorem | vonmea 46589 | (voln‘𝑋) is a measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set 𝑋. Comments in Definition 115E of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln‘𝑋) ∈ Meas) | ||
| Theorem | volicon0 46590 | The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (vol‘(𝐴[,)𝐵)) = (𝐵 − 𝐴)) | ||
| Theorem | hsphoif 46591* | 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) ⇒ ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ) | ||
| Theorem | hoidmvval 46592* | The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) & ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))) | ||
| Theorem | hoissrrn2 46593* | A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑m 𝑋)) | ||
| Theorem | hsphoival 46594* | 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) & ⊢ (𝜑 → 𝐾 ∈ 𝑋) ⇒ ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))) | ||
| Theorem | hoiprodcl3 46595* | The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,)𝐵)) ∈ (0[,)+∞)) | ||
| Theorem | volicore 46596 | The Lebesgue measure of a left-closed right-open interval is a real number. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ) | ||
| Theorem | hoidmvcl 46597* | The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) ⇒ ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞)) | ||
| Theorem | hoidmv0val 46598* | The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) & ⊢ (𝜑 → 𝐴:∅⟶ℝ) & ⊢ (𝜑 → 𝐵:∅⟶ℝ) ⇒ ⊢ (𝜑 → (𝐴(𝐿‘∅)𝐵) = 0) | ||
| Theorem | hoidmvn0val 46599* | The dimensional volume of a non-0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) ⇒ ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) | ||
| Theorem | hsphoidmvle2 46600* | The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌)) & ⊢ 𝑋 = (𝑌 ∪ {𝑍}) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) & ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) & ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) & ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) ⇒ ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵))) | ||
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