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Theorem List for Metamath Proof Explorer - 41101-41200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremovn0 41101 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 41102 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 41103 For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0)

Theoremovnxrcl 41104 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 41105* 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 41106* (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 41107* (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 41108 (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 41109 (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 41110 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 41111* 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 41112* 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 41113* 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 41114* 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 41115* 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 41116 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 41117* 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 41118* 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 41119* 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 41120* 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 41121* 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 41122* 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 41123* 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 41124* 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 41125* 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 41126* 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 41127* 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 41128* 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 non-empty, 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 41129* 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 41130* 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 41131* 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 41132* 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 41133* 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 41134* 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 41135* 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 41136* 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 41137* 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 41138* 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 41139 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 41140* 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 41141 Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))

Theoremhoi2toco 41142* 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 41143* 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 41144* 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 41145* 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 41146* 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 41147 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 41148 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 41149 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 41150* 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 41151* 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 41152 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 41153* 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 41154* 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 41155* 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 41156 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 41157* 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)))

Theoremhoiqssbllem2 41158* 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)))))))       (𝜑X𝑖𝑋 ((𝐶𝑖)[,)(𝐷𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))

Theoremhoiqssbllem3 41159* A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑐 ∈ (ℚ ↑𝑚 𝑋)∃𝑑 ∈ (ℚ ↑𝑚 𝑋)(𝑌X𝑖𝑋 ((𝑐𝑖)[,)(𝑑𝑖)) ∧ X𝑖𝑋 ((𝑐𝑖)[,)(𝑑𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Theoremhoiqssbl 41160* A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑐 ∈ (ℚ ↑𝑚 𝑋)∃𝑑 ∈ (ℚ ↑𝑚 𝑋)(𝑌X𝑖𝑋 ((𝑐𝑖)[,)(𝑑𝑖)) ∧ X𝑖𝑋 ((𝑐𝑖)[,)(𝑑𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Theoremhspmbllem1 41161* Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (a) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐾𝑋)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( ∈ (𝑋 ∖ {𝐾}), (𝑐), if((𝑐) ≤ 𝑦, (𝑐), 𝑦)))))    &   𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( = 𝐾, if(𝑥 ≤ (𝑐), (𝑐), 𝑥), (𝑐)))))       (𝜑 → (𝐴(𝐿𝑋)𝐵) = ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) +𝑒 (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)))

Theoremhspmbllem2 41162* Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (b) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐾𝑋)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))    &   (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ)    &   (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ)    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( ∈ (𝑋 ∖ {𝐾}), (𝑐), if((𝑐) ≤ 𝑦, (𝑐), 𝑦)))))    &   𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( = 𝐾, if(𝑥 ≤ (𝑐), (𝑐), 𝑥), (𝑐)))))       (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))

Theoremhspmbllem3 41163* Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. This proof handles the non-trivial cases (nonzero dimension and finite outer measure) (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐾𝑋)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   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)}))    &   𝐵 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑘))))    &   𝑇 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑘))))       (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))

Theoremhspmbl 41164* Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐾𝑋)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))

Theoremhoimbllem 41165* Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑖𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))       (𝜑X𝑖𝑋 ((𝐴𝑖)[,)(𝐵𝑖)) ∈ 𝑆)

Theoremhoimbl 41166* Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)[,)(𝐵𝑖)) ∈ 𝑆)

Theoremopnvonmbllem1 41167* The half-open interval expressed using a composition of a function (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑖𝜑    &   (𝜑𝑋𝑉)    &   (𝜑𝐶:𝑋⟶ℚ)    &   (𝜑𝐷:𝑋⟶ℚ)    &   (𝜑X𝑖𝑋 ((𝐶𝑖)[,)(𝐷𝑖)) ⊆ 𝐵)    &   (𝜑𝐵𝐺)    &   (𝜑𝑌X𝑖𝑋 ((𝐶𝑖)[,)(𝐷𝑖)))    &   𝐾 = { ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∣ X𝑖𝑋 (([,) ∘ )‘𝑖) ⊆ 𝐺}    &   𝐻 = (𝑖𝑋 ↦ ⟨(𝐶𝑖), (𝐷𝑖)⟩)       (𝜑 → ∃𝐾 𝑌X𝑖𝑋 (([,) ∘ )‘𝑖))

Theoremopnvonmbllem2 41168* An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))    &   𝐾 = { ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∣ X𝑖𝑋 (([,) ∘ )‘𝑖) ⊆ 𝐺}       (𝜑𝐺𝑆)

Theoremopnvonmbl 41169 An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))       (𝜑𝐺𝑆)

Theoremopnssborel 41170 Open sets of a generalized real Euclidean space are Borel sets (notice that this theorem is even more general, because 𝑋 is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (TopOpen‘(ℝ^‘𝑋))    &   𝐵 = (SalGen‘𝐴)       𝐴𝐵

Theoremborelmbl 41171 All Borel subsets of the n-dimensional Real numbers are Lebesgue measurable. This is Proposition 115G (b) of [Fremlin1] p. 32. See also Definition 111G (d) of [Fremlin1] p. 13. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))       (𝜑𝐵𝑆)

Theoremvolicorege0 41172 The Lebesgue measure of a left-closed right-open interval with real bounds, is a nonnegative real number. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,)+∞))

Theoremisvonmbl 41173* The predicate "𝐴 is measurable w.r.t. the n-dimensional Lebesgue measure". A set is measurable if it splits every other set 𝑥 in a "nice" way, that is, if the measure of the pieces 𝑥𝐴 and 𝑥𝐴 sum up to the measure of 𝑥. Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ (𝐸 ⊆ (ℝ ↑𝑚 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)(((voln*‘𝑋)‘(𝑎𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎𝐸))) = ((voln*‘𝑋)‘𝑎))))

Theoremmblvon 41174 The n-dimensional Lebesgue measure of a measurable set is the same as its n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ dom (voln‘𝑋))       (𝜑 → ((voln‘𝑋)‘𝐴) = ((voln*‘𝑋)‘𝐴))

Theoremvonmblss 41175 n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)       (𝜑 → dom (voln‘𝑋) ⊆ 𝒫 (ℝ ↑𝑚 𝑋))

Theoremvolico2 41176 The measure of left closed, right open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴𝐵, (𝐵𝐴), 0))

Theoremvonmblss2 41177 n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ dom (voln‘𝑋))       (𝜑𝑌 ⊆ (ℝ ↑𝑚 𝑋))

Theoremovolval2lem 41178* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))       (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))

Theoremovolval2 41179* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. See ovolval 23288 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovnsubadd2lem 41180* (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))    &   𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))       (𝜑 → ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Theoremovnsubadd2 41181 (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Theoremovolval3 41182* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23288 and ovolval2 41179 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovnsplit 41183 The n-dimensional Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (((voln*‘𝑋)‘(𝐴𝐵)) +𝑒 ((voln*‘𝑋)‘(𝐴𝐵))))

Theoremovolval4lem1 41184* |- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:ℕ⟶(ℝ* × ℝ*))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)    &   𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))}       (𝜑 → ( ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

Theoremovolval4lem2 41185* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41182, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovolval4 41186* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41182, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovolval5lem1 41187* |- ( ph -> ( sum^ (𝑛 ∈ ℕ ↦ (vol ( ( A - ( W / ( 2 ^ n ) ) ) (,) B ) ) ) ) <_ ( ( sum^ (𝑛 ∈ ℕ ↦ (vol ( A [,) B ) ) ) ) +e W ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ+)    &   𝐶 = {𝑛 ∈ ℕ ∣ 𝐴 < 𝐵}       (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +𝑒 𝑊))

Theoremovolval5lem2 41188* |- ( ( ph /\ n e. NN ) -> <. ( ( 1st (𝐹 n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd (𝐹 n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   (𝜑𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    &   𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))    &   (𝜑𝐹:ℕ⟶(ℝ × ℝ))    &   (𝜑𝐴 ran ([,) ∘ 𝐹))    &   (𝜑𝑊 ∈ ℝ+)    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)       (𝜑 → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))

Theoremovolval5lem3 41189* The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    &   𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}       inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Theoremovolval5 41190* The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovnovollem1 41191* if 𝐹 is a cover of 𝐵 in , then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))    &   𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})    &   (𝜑𝐵 ran ([,) ∘ 𝐹))    &   (𝜑𝐵𝑊)    &   (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))       (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))

Theoremovnovollem2 41192* if 𝐼 is a cover of (𝐵𝑚 {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))    &   (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))    &   (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))    &   𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))       (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))

Theoremovnovollem3 41193* The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}       (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))

Theoremovnovol 41194 The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)       (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))

Theoremvonvolmbllem 41195* If a subset 𝐵 of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with 𝑛 equal to 1). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦𝐵)) +𝑒 (vol*‘(𝑦𝐵))))    &   (𝜑𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵𝑚 {𝐴})))) = ((voln*‘{𝐴})‘𝑋))

Theoremvonvolmbl 41196 A subset of Real numbers is Lebesgue measurable if and only if its corresponding 1-dimensional set is measurable w.r.t. the 1-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)       (𝜑 → ((𝐵𝑚 {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Theoremvonvol 41197 The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ dom vol)       (𝜑 → ((voln‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol‘𝐵))

Theoremvonvolmbl2 41198* A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection 𝑌 on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝑌    &   (𝜑𝐴𝑉)    &   (𝜑𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol))

Theoremvonvol2 41199* The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝑌    &   (𝜑𝐴𝑉)    &   (𝜑𝑋 ∈ dom (voln‘{𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌))

Theoremhoimbl2 41200* Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑X𝑘𝑋 (𝐴[,)𝐵) ∈ 𝑆)

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