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Theorem List for Metamath Proof Explorer - 41001-41100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmeadjiun 41001* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≼ ω)    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → (𝑀 𝑘𝐴 𝐵) = (Σ^‘(𝑘𝐴 ↦ (𝑀𝐵))))
 
Theoremismeannd 41002* Sufficient condition to prove that 𝑀 is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝑀:𝑆⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑒:ℕ⟶𝑆Disj 𝑛 ∈ ℕ (𝑒𝑛)) → (𝑀 𝑛 ∈ ℕ (𝑒𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒𝑛)))))       (𝜑𝑀 ∈ Meas)
 
Theoremmeaiunlelem 41003* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑆)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑 → (𝑀 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))))
 
Theoremmeaiunle 41004* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑆)       (𝜑 → (𝑀 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))))
 
Theorempsmeasurelem 41005* 𝑀 applied to a disjoint union of subsets of its domain is the sum of 𝑀 applied to such subset. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐻:𝑋⟶(0[,]+∞))    &   𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻𝑥)))    &   (𝜑𝑀:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   (𝜑Disj 𝑦𝑌 𝑦)       (𝜑 → (𝑀 𝑌) = (Σ^‘(𝑀𝑌)))
 
Theorempsmeasure 41006* Point supported measure, Remark 112B (d) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐻:𝑋⟶(0[,]+∞))    &   𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻𝑥)))       (𝜑𝑀 ∈ Meas)
 
Theoremvoliunsge0lem 41007* The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑆 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))    &   (𝜑𝐸:ℕ⟶dom vol)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))       (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
 
Theoremvoliunsge0 41008* The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐸:ℕ⟶dom vol)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))       (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
 
Theoremvolmea 41009 The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑 → vol ∈ Meas)
 
Theoremmeage0 41010 If the measure of a measurable set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)       (𝜑 → 0 ≤ (𝑀𝐴))
 
Theoremmeadjunre 41011 The measure of the union of two disjoint sets, with finite measure, is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑 → (𝑀𝐵) ∈ ℝ)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) + (𝑀𝐵)))
 
Theoremmeassre 41012 If the measure of a measurable set is real, then the measure of any of its measurable subsets is real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ∈ dom 𝑀)       (𝜑 → (𝑀𝐵) ∈ ℝ)
 
Theoremmeale0eq0 41013 A measure that is smaller or equal to 0 is 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ≤ 0)       (𝜑 → (𝑀𝐴) = 0)
 
Theoremmeadif 41014 The measure of the difference of two sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) − (𝑀𝐵)))
 
Theoremmeaiuninclem 41015* Measures are continuous from below (bounded case): if 𝐸 is a sequence of increasing measurable sets (with uniformly bounded measure) then the measure of the union is the union of the measure. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 (𝑀‘(𝐸𝑛)) ≤ 𝑥)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiuninc 41016* Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 (𝑀‘(𝐸𝑛)) ≤ 𝑥)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiuninc2 41017* Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑𝐵 ∈ ℝ)    &   ((𝜑𝑛𝑍) → (𝑀‘(𝐸𝑛)) ≤ 𝐵)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiunincf 41018* Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
𝑛𝜑    &   𝑛𝐸    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 (𝑀‘(𝐸𝑛)) ≤ 𝑥)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiuninc3v 41019* Measures are continuous from below: if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 41016 and meaiuninc2 41017 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆~~>*(𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiuninc3 41020* Measures are continuous from below: if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 41016 and meaiuninc2 41017 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
𝑛𝜑    &   𝑛𝐸    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆~~>*(𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiininclem 41021* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑𝐾 ∈ (ℤ𝑁))    &   (𝜑 → (𝑀‘(𝐸𝐾)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))    &   𝐺 = (𝑛𝑍 ↦ ((𝐸𝐾) ∖ (𝐸𝑛)))    &   𝐹 = 𝑛𝑍 (𝐺𝑛)       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiininc 41022* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑𝐾 ∈ (ℤ𝑁))    &   (𝜑 → (𝑀‘(𝐸𝐾)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
Theoremmeaiininc2 41023* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
𝑛𝜑    &   𝑘𝜑    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑 → ∃𝑘𝑍 (𝑀‘(𝐸𝑘)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))
 
20.32.19.4  Outer measures and Caratheodory's construction

Proofs for most of the theorems in section 113 of [Fremlin1]

 
Syntaxcome 41024 Extend class notation with the class of outer measures.
class OutMeas
 
Definitiondf-ome 41025* 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 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))))}
 
Syntaxccaragen 41026 Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
class CaraGen
 
Definitiondf-caragen 41027* 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‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
 
Theoremcaragenval 41028* 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‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
 
Theoremisome 41029* Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑂𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
 
Theoremcaragenel 41030* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
 
Theoremomef 41031 An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂       (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
 
Theoremome0 41032 The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)       (𝜑 → (𝑂‘∅) = 0)
 
Theoremomessle 41033 The outer measure of a set is larger or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐵𝑋)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))
 
Theoremomedm 41034 The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
 
Theoremcaragensplit 41035 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐸𝑆)    &   (𝜑𝐴𝑋)       (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
 
Theoremcaragenelss 41036 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐴𝑆)    &   𝑋 = dom 𝑂       (𝜑𝐴𝑋)
 
Theoremcarageneld 41037* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸 ∈ 𝒫 𝑋)    &   ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))       (𝜑𝐸𝑆)
 
Theoremomecl 41038 The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
 
Theoremcaragenss 41039 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)
 
Theoremomeunile 41040 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   (𝜑𝑌 ≼ ω)       (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
 
Theoremcaragen0 41041 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → ∅ ∈ 𝑆)
 
Theoremomexrcl 41042 The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ∈ ℝ*)
 
Theoremcaragenunidm 41043 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑋𝑆)
 
Theoremcaragensspw 41044 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑆 ⊆ 𝒫 𝑋)
 
Theoremomessre 41045 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑂𝐵) ∈ ℝ)
 
Theoremcaragenuni 41046 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 𝑆 = dom 𝑂)
 
Theoremcaragenuncllem 41047 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸𝐹)))) = (𝑂𝐴))
 
Theoremcaragenuncl 41048 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)
 
Theoremcaragendifcl 41049 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)       (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
 
Theoremcaragenfiiuncl 41050* The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)       (𝜑 𝑘𝐴 𝐵𝑆)
 
Theoremomeunle 41051 The outer measure of the union of two sets is less or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝑂‘(𝐴𝐵)) ≤ ((𝑂𝐴) +𝑒 (𝑂𝐵)))
 
Theoremomeiunle 41052* 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.)
𝑛𝜑    &   𝑛𝐸    &   (𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶𝒫 𝑋)       (𝜑 → (𝑂 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremomelesplit 41053 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ≤ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
 
Theoremomeiunltfirp 41054* 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶𝒫 𝑋)    &   (𝜑 → (𝑂 𝑛𝑍 (𝐸𝑛)) ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ+)       (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂 𝑛𝑍 (𝐸𝑛)) < (Σ𝑛𝑧 (𝑂‘(𝐸𝑛)) + 𝑌))
 
Theoremomeiunlempt 41055* 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.)
𝑛𝜑    &   (𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   ((𝜑𝑛𝑍) → 𝐸𝑋)       (𝜑 → (𝑂 𝑛𝑍 𝐸) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑂𝐸))))
 
Theoremcarageniuncllem1 41056* The outer measure of 𝐴 ∩ (𝐺𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹𝑚). 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))    &   (𝜑𝐾𝑍)       (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
 
Theoremcarageniuncllem2 41057* 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   (𝜑𝑌 ∈ ℝ+)    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))       (𝜑 → ((𝑂‘(𝐴 𝑛𝑍 (𝐸𝑛))) +𝑒 (𝑂‘(𝐴 𝑛𝑍 (𝐸𝑛)))) ≤ ((𝑂𝐴) + 𝑌))
 
Theoremcarageniuncl 41058* 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)       (𝜑 𝑛𝑍 (𝐸𝑛) ∈ 𝑆)
 
Theoremcaragenunicl 41059 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝑋𝑆)    &   (𝜑𝑋 ≼ ω)       (𝜑 𝑋𝑆)
 
Theoremcaragensal 41060 Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑆 ∈ SAlg)
 
Theoremcaratheodorylem1 41061* 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   (𝜑Disj 𝑛𝑍 (𝐸𝑛))    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremcaratheodorylem2 41062* 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸:ℕ⟶𝑆)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))    &   𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))       (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremcaratheodory 41063 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → (𝑂𝑆) ∈ Meas)
 
Theorem0ome 41064* The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)    &   𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)       (𝜑𝑂 ∈ OutMeas)
 
Theoremisomenndlem 41065* 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.)
(𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑 → (𝑂‘∅) = 0)    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))    &   (𝜑𝐵 ⊆ ℕ)    &   (𝜑𝐹:𝐵1-1-onto𝑌)    &   𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))       (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
 
Theoremisomennd 41066* Sufficient condition to prove that 𝑂 is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑 → (𝑂‘∅) = 0)    &   ((𝜑𝑥𝑋𝑦𝑥) → (𝑂𝑦) ≤ (𝑂𝑥))    &   ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))       (𝜑𝑂 ∈ OutMeas)
 
Theoremcaragenel2d 41067* Membership in the Caratheodory's construction. Similar to carageneld 41037, but here "less then or equal" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸 ∈ 𝒫 𝑋)    &   ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) ≤ (𝑂𝑎))       (𝜑𝐸𝑆)
 
Theoremomege0 41068 If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → 0 ≤ (𝑂𝐴))
 
Theoremomess0 41069 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) = 0)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑂𝐵) = 0)
 
Theoremcaragencmpl 41070 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐸𝑋)    &   (𝜑 → (𝑂𝐸) = 0)    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝐸𝑆)
 
20.32.19.5  Lebesgue measure on n-dimensional Real numbers

Proofs for most of the theorems in section 115 of [Fremlin1]

 
Syntaxcovoln 41071 Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
class voln*
 
Definitiondf-ovoln 41072* 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 X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
 
Syntaxcvoln 41073 Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
class voln
 
Definitiondf-voln 41074 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))))
 
Theoremvonval 41075 Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
 
Theoremovnval 41076* Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ 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 X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
 
Theoremelhoi 41077* Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)       (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑𝑚 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥𝑋 (𝑌𝑥) ∈ (𝐴[,)𝐵))))
 
Theoremicoresmbl 41078 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
 
Theoremhoissre 41079* The projection of a half-open interval onto a single dimension is a subset of . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐼:𝑋⟶(ℝ × ℝ))       ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)
 
Theoremovnval2 41080* Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
 
Theoremvolicorecl 41081 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)
 
Theoremhoiprodcl 41082* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞))
 
Theoremhoicvr 41083* 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 ↦ (𝑥𝑋 ↦ ⟨-𝑗, 𝑗⟩))    &   (𝜑𝑋 ∈ Fin)       (𝜑 → (ℝ ↑𝑚 𝑋) ⊆ 𝑗 ∈ ℕ X𝑖𝑋 (([,) ∘ (𝐼𝑗))‘𝑖))
 
Theoremhoissrrn 41084* A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
 
Theoremovn0val 41085 The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ⊆ (ℝ ↑𝑚 ∅))       (𝜑 → ((voln*‘∅)‘𝐴) = 0)
 
Theoremovnn0val 41086* The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < ))
 
Theoremovnval2b 41087* Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})       (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf((𝐿𝐴), ℝ*, < )))
 
Theoremvolicorescl 41088 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)
 
Theoremovnprodcl 41089* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   (𝜑𝐼 ∈ ℕ)       (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐹𝐼))‘𝑘)) ∈ (0[,)+∞))
 
Theoremhoiprodcl2 41090* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   (𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑 → (𝐿𝐼) ∈ (0[,)+∞))
 
Theoremhoicvrrex 41091* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
 
Theoremovnsupge0 41092* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑𝑀 ⊆ (0[,]+∞))
 
Theoremovnlecvr 41093* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
 
Theoremovnpnfelsup 41094* +∞ is an element of the set used in the definition of the Lebesgue outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → +∞ ∈ 𝑀)
 
Theoremovnsslelem 41095* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐵 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵))
 
Theoremovnssle 41096 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵))
 
Theoremovnlerp 41097* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ∃𝑧𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
 
Theoremovnf 41098 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.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
 
Theoremovncvrrp 41099* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})    &   𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))    &   𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))       (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷𝐴)‘𝐸))
 
Theoremovn0lem 41100* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))}    &   (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))    &   𝐼 = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨1, 0⟩))       (𝜑 → inf(𝑀, ℝ*, < ) = 0)
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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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