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Theorem List for Metamath Proof Explorer - 43101-43200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcarageniuncl 43101* 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 43102 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 43103 Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑆 ∈ SAlg)

Theoremcaratheodorylem1 43104* 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 43105* 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 43106 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 43107* 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 43108* 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 43109* 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 43110* Membership in the Caratheodory's construction. Similar to carageneld 43080, 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸 ∈ 𝒫 𝑋)    &   ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) ≤ (𝑂𝑎))       (𝜑𝐸𝑆)

Theoremomege0 43111 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 43112 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 43113 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)

20.37.19.5  Lebesgue measure on n-dimensional Real numbers

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

Syntaxcovoln 43114 Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
class voln*

Definitiondf-ovoln 43115* 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 43116 Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
class voln

Definitiondf-voln 43117 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 43118 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 43119* 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 43120* 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 43121 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 43122* 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 43123* 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 43124 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 43125* 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 43126* 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 43127* 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 43128 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 43129* 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 43130* 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 43131 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 43132* 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 43133* 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 43134* 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 43135* 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 43136* 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 43137* +∞ 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 43138* 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 43139 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 43140* 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 43141 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 43142* 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 43143* 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)

Theoremovn0 43144 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.)
(𝜑𝑋 ∈ Fin)       (𝜑 → ((voln*‘𝑋)‘∅) = 0)

Theoremovncl 43145 The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ (0[,]+∞))

Theoremovn02 43146 For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0)

Theoremovnxrcl 43147 The Lebesgue outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ*)

Theoremovnsubaddlem1 43148* The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑍 = (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)))))})    &   𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ { ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘)})    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))    &   ((𝜑𝑛 ∈ ℕ) → (𝐼𝑛) ∈ ((𝐷‘(𝐴𝑛))‘(𝐸 / (2↑𝑛))))    &   (𝜑𝐹:ℕ–1-1-onto→(ℕ × ℕ))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1st ‘(𝐹𝑚)))‘(2nd ‘(𝐹𝑚))))       (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝐸))

Theoremovnsubaddlem2 43149* (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑍 = (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)))))})    &   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 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝐸))

Theoremovnsubadd 43150* (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))       (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))

Theoremovnome 43151 (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.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln*‘𝑋) ∈ OutMeas)

Theoremvonmea 43152 (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.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln‘𝑋) ∈ Meas)

Theoremvolicon0 43153 The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (vol‘(𝐴[,)𝐵)) = (𝐵𝐴))

Theoremhsphoif 43154* 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) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)

Theoremhoidmvval 43155* 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ∈ Fin)       (𝜑 → (𝐴(𝐿𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))))

Theoremhoissrrn2 43156* A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)       (𝜑X𝑘𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑m 𝑋))

Theoremhsphoival 43157* 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) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐾𝑋)       (𝜑 → (((𝐻𝐴)‘𝐵)‘𝐾) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))

Theoremhoiprodcl3 43158* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝑋 (vol‘(𝐴[,)𝐵)) ∈ (0[,)+∞))

Theoremvolicore 43159 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)

Theoremhoidmvcl 43160* 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))

Theoremhoidmv0val 43161* 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝐴:∅⟶ℝ)    &   (𝜑𝐵:∅⟶ℝ)       (𝜑 → (𝐴(𝐿‘∅)𝐵) = 0)

Theoremhoidmvn0val 43162* 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremhsphoidmvle2 43163* 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶𝐷)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))

Theoremhsphoidmvle 43164* The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)𝐵))

Theoremhoidmvval0 43165* The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑗𝜑    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑 → ∃𝑗𝑋 (𝐵𝑗) ≤ (𝐴𝑗))       (𝜑 → (𝐴(𝐿𝑋)𝐵) = 0)

Theoremhoiprodp1 43166* The dimensional volume of a half-open interval with dimension 𝑛 + 1. Used in the first equality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑌 ∈ Fin)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍𝑌)    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐺 = ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) = (𝐺 · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))))

Theoremsge0hsphoire 43167* If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑌 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑊𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝑆 ∈ ℝ)       (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)

Theoremhoidmvval0b 43168* The dimensional volume of the (half-open interval) empty set. 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐴) = 0)

Theoremhoidmv1lelem1 43169* The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   𝑆 = sup(𝑈, ℝ, < )       (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))

Theoremhoidmv1lelem2 43170* This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆 < 𝐵)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑆 ∈ ((𝐶𝐾)[,)(𝐷𝐾)))    &   𝑀 = if((𝐷𝐾) ≤ 𝐵, (𝐷𝐾), 𝐵)       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)

Theoremhoidmv1lelem3 43171* The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the nonempty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   𝑆 = sup(𝑈, ℝ, < )       (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))

Theoremhoidmv1le 43172* The dimensional volume of a 1-dimensional half-open interval is less than or equal to the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is one of the two base cases of the induction of Lemma 115B of [Fremlin1] p. 29 (the other base case is the 0-dimensional case). This proof of the 1-dimensional case is given in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑍𝑉)    &   𝑋 = {𝑍}    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))

Theoremhoidmvlelem1 43173* The supremum of 𝑈 belongs to 𝑈. Step (c) in the proof 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   𝑆 = sup(𝑈, ℝ, < )    &   (𝜑 → (𝐴𝑍) < (𝐵𝑍))       (𝜑𝑆𝑈)

Theoremhoidmvlelem2 43174* This is the contradiction proven in step (d) in the proof 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   𝐹 = (𝑦𝑌 ↦ 0)    &   𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝑆 < (𝐵𝑍))    &   𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)))    &   𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))    &   𝑉 = ({(𝐵𝑍)} ∪ 𝑂)    &   𝑄 = inf(𝑉, ℝ, < )       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)

Theoremhoidmvlelem3 43175* This is the contradiction proven in step (d) in the proof 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))    &   𝐹 = (𝑦𝑌 ↦ 0)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝑆 < (𝐵𝑍))    &   𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   𝑂 = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)

Theoremhoidmvlelem4 43176* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑌 ≠ ∅)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   𝑆 = sup(𝑈, ℝ, < )    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))

Theoremhoidmvlelem5 43177* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   (𝜑𝑌 ≠ ∅)       (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))

Theoremhoidmvle 43178* The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))

Theoremovnhoilem1 43179* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))       (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremovnhoilem2 43180* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))    &   𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Theoremovnhoi 43181* The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))

Theoremdmovn 43182 The domain of the Lebesgue outer measure is the power set of the n-dimensional Real numbers. Step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))

Theoremhoicoto2 43183* The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼:𝑋⟶(ℝ × ℝ))    &   𝐴 = (𝑘𝑋 ↦ (1st ‘(𝐼𝑘)))    &   𝐵 = (𝑘𝑋 ↦ (2nd ‘(𝐼𝑘)))       (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))

Theoremdmvon 43184 Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))

Theoremhoi2toco 43185* The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑘𝜑    &   𝐼 = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)       (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))

Theoremhoidifhspval 43186* 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → (𝐷𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌), (𝑎𝑘)))))

Theoremhspval 43187* The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (𝑥 ∈ Fin ↦ (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐼𝑋)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

Theoremovnlecvr2 43188* 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. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))    &   (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))

Theoremovncvr2 43189* 𝐵 and 𝑇 are the left and right side of a cover of 𝐴. This cover is made of n-dimensional half-open intervals and approximates the n-dimensional Lebesgue outer volume of 𝐴. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (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) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))    &   (𝜑𝐼 ∈ ((𝐷𝐴)‘𝐸))    &   𝐵 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))    &   𝑇 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))       (𝜑 → (((𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))

Theoremdmovnsal 43190 The domain of the Lebesgue measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)       (𝜑𝑆 ∈ SAlg)

Theoremunidmovn 43191 Base set of the n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 dom (voln*‘𝑋) = (ℝ ↑m 𝑋))

Theoremrrnmbl 43192 The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋))

Theoremhoidifhspval2 43193* 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴:𝑋⟶ℝ)       (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))

Theoremhspdifhsp 43194* A n-dimensional half-open interval is the intersection of the difference of half spaces. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑖𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))       (𝜑X𝑖𝑋 ((𝐴𝑖)[,)(𝐵𝑖)) = 𝑖𝑋 ((𝑖(𝐻𝑋)(𝐵𝑖)) ∖ (𝑖(𝐻𝑋)(𝐴𝑖))))

Theoremunidmvon 43195 Base set of the n-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)       (𝜑 𝑆 = (ℝ ↑m 𝑋))

Theoremhoidifhspf 43196* 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴:𝑋⟶ℝ)       (𝜑 → ((𝐷𝑌)‘𝐴):𝑋⟶ℝ)

Theoremhoidifhspval3 43197* 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐽𝑋)       (𝜑 → (((𝐷𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))

Theoremhoidifhspdmvle 43198* The dimensional volume of the difference of a half-open interval and a half-space is less than or equal to the dimensional volume of the whole half-open interval. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐾𝑋)    &   𝐷 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑋 ↦ if( = 𝐾, if(𝑥 ≤ (𝑐), (𝑐), 𝑥), (𝑐)))))    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → (((𝐷𝑌)‘𝐴)(𝐿𝑋)𝐵) ≤ (𝐴(𝐿𝑋)𝐵))

Theoremvoncmpl 43199 The Lebesgue measure is complete. See Definition 112Df of [Fremlin1] p. 19. This is an observation written after Definition 115E of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐸 ∈ dom (voln‘𝑋))    &   (𝜑 → ((voln‘𝑋)‘𝐸) = 0)    &   (𝜑𝐹𝐸)       (𝜑𝐹𝑆)

Theoremhoiqssbllem1 43200* The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑖𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝑌 ∈ (ℝ ↑m 𝑋))    &   (𝜑𝐶:𝑋⟶ℝ)    &   (𝜑𝐷:𝑋⟶ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   ((𝜑𝑖𝑋) → (𝐶𝑖) ∈ (((𝑌𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌𝑖)))    &   ((𝜑𝑖𝑋) → (𝐷𝑖) ∈ ((𝑌𝑖)(,)((𝑌𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))       (𝜑𝑌X𝑖𝑋 ((𝐶𝑖)[,)(𝐷𝑖)))

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