P. 467 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46601-46700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ovnhoi 46601*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	dmovn 46602	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
Theorem	hoicoto2 46603*	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.)
⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))    &   ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))    &   ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
Theorem	dmvon 46604	Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
Theorem	hoi2toco 46605*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
Theorem	hoidifhspval 46606*	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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
Theorem	hspval 46607*	The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Theorem	ovnlecvr2 46608*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
Theorem	ovncvr2 46609*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})    &   ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))    &   ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))    &   ⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸))    &   ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))    &   ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))    ⇒   ⊢ (𝜑 → (((𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
Theorem	dmovnsal 46610	The domain of the Lebesgue measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	unidmovn 46611	Base set of the n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋))
Theorem	rrnmbl 46612	The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋))
Theorem	hoidifhspval2 46613*	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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
Theorem	hspdifhsp 46614*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = ∩ 𝑖 ∈ 𝑋 ((𝑖(𝐻‘𝑋)(𝐵‘𝑖)) ∖ (𝑖(𝐻‘𝑋)(𝐴‘𝑖))))
Theorem	unidmvon 46615	Base set of the n-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    ⇒   ⊢ (𝜑 → ∪ 𝑆 = (ℝ ↑m 𝑋))
Theorem	hoidifhspf 46616*	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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ)
Theorem	hoidifhspval3 46617*	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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))
Theorem	hoidifhspdmvle 46618*	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, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) ≤ (𝐴(𝐿‘𝑋)𝐵))
Theorem	voncmpl 46619	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋))    &   ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	hoiqssbllem1 46620*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))    ⇒   ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
Theorem	hoiqssbllem2 46621*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
Theorem	hoiqssbllem3 46622*	A n-dimensional ball contains a nonempty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑m 𝑋)∃𝑑 ∈ (ℚ ↑m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
Theorem	hoiqssbl 46623*	A n-dimensional ball contains a nonempty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑m 𝑋)∃𝑑 ∈ (ℚ ↑m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
Theorem	hspmbllem1 46624*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))    &   ⊢ 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((𝐴(𝐿‘𝑋)((𝑇‘𝑌)‘𝐵)) +𝑒 (((𝑆‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵)))
Theorem	hspmbllem2 46625*	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 ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))    &   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))    &   ⊢ 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    ⇒   ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))
Theorem	hspmbllem3 46626*	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 ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})    &   ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))    &   ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))    &   ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑘))))    &   ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑘))))    ⇒   ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))
Theorem	hspmbl 46627*	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 ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))
Theorem	hoimbllem 46628*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	hoimbl 46629*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	opnvonmbllem1 46630*	The half-open interval expressed using a composition of a function (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℚ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℚ)    &   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))    &   ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}    &   ⊢ 𝐻 = (𝑖 ∈ 𝑋 ↦ ⟨(𝐶‘𝑖), (𝐷‘𝑖)⟩)    ⇒   ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
Theorem	opnvonmbllem2 46631*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))    &   ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑆)
Theorem	opnvonmbl 46632	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑆)
Theorem	opnssborel 46633	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)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	borelmbl 46634	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝑆)
Theorem	volicorege0 46635	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[,)+∞))
Theorem	isvonmbl 46636*	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 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 sum up to the measure of 𝑥. Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ (𝐸 ⊆ (ℝ ↑m 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎))))
Theorem	mblvon 46637	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = ((voln*‘𝑋)‘𝐴))
Theorem	vonmblss 46638	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln‘𝑋) ⊆ 𝒫 (ℝ ↑m 𝑋))
Theorem	volico2 46639	The measure of left-closed right-open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 ≤ 𝐵, (𝐵 − 𝐴), 0))
Theorem	vonmblss2 46640	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ dom (voln‘𝑋))    ⇒   ⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑m 𝑋))
Theorem	ovolval2lem 46641*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    ⇒   ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
Theorem	ovolval2 46642*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. See ovolval 25374 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnsubadd2lem 46643*	(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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
Theorem	ovnsubadd2 46644	(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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
Theorem	ovolval3 46645*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 25374 and ovolval2 46642 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnsplit 46646	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (((voln*‘𝑋)‘(𝐴 ∩ 𝐵)) +𝑒 ((voln*‘𝑋)‘(𝐴 ∖ 𝐵))))
Theorem	ovolval4lem1 46647*	|- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)    &   ⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))}    ⇒   ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
Theorem	ovolval4lem2 46648*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 46645, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓‘𝑛)), if((1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛)), (1st ‘(𝑓‘𝑛)))⟩)    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovolval4 46649*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 46645, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovolval5lem1 46650*	⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛) ))(,)𝐵)))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵) ))) +𝑒 𝑊)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑊 ∈ ℝ+)    &   ⊢ 𝐶 = {𝑛 ∈ ℕ ∣ 𝐴 < 𝐵}    ⇒   ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +𝑒 𝑊))
Theorem	ovolval5lem2 46651*	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩ ∈ (ℝ × ℝ)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   ⊢ (𝜑 → 𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    &   ⊢ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,) ∘ 𝐹))    &   ⊢ (𝜑 → 𝑊 ∈ ℝ+)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
Theorem	ovolval5lem3 46652*	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*, < )
Theorem	ovolval5 46653*	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.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnovollem1 46654*	if 𝐹 is a cover of 𝐵 in ℝ, then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))    &   ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹‘𝑗)⟩})    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
Theorem	ovnovollem2 46655*	if 𝐼 is a cover of (𝐵 ↑m {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))    &   ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))    &   ⊢ (𝜑 → 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))    &   ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
Theorem	ovnovollem3 46656*	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}    &   ⊢ 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))
Theorem	ovnovol 46657	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))
Theorem	vonvolmbllem 46658*	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))))    &   ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴})))) = ((voln*‘{𝐴})‘𝑋))
Theorem	vonvolmbl 46659	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((𝐵 ↑m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))
Theorem	vonvol 46660	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ dom vol)    ⇒   ⊢ (𝜑 → ((voln‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol‘𝐵))
Theorem	vonvolmbl2 46661*	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.)
⊢ Ⅎ𝑓𝑌    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol))
Theorem	vonvol2 46662*	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑓𝑌    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ dom (voln‘{𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌))
Theorem	hoimbl2 46663*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆)
Theorem	voncl 46664	The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) ∈ (0[,]+∞))
Theorem	vonhoi 46665*	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). A direct consequence of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	vonxrcl 46666	The Lebesgue measure of a set is an extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) ∈ ℝ*)
Theorem	ioosshoi 46667	A n-dimensional open interval is a subset of the half-open interval with the same bounds. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ X𝑘 ∈ 𝑋 (𝐴[,)𝐵)
Theorem	vonn0hoi 46668*	The Lebesgue outer measure of a multidimensional half-open interval when the dimension of the space is nonzero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
Theorem	von0val 46669	The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ dom (voln‘∅))    ⇒   ⊢ (𝜑 → ((voln‘∅)‘𝐴) = 0)
Theorem	vonhoire 46670*	The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (𝐴[,)𝐵)) ∈ ℝ)
Theorem	iinhoiicclem 46671*	A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))))    ⇒   ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
Theorem	iinhoiicc 46672*	A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
Theorem	iunhoiioolem 46673*	A n-dimensional open interval expressed as the indexed union of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))    &   ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
Theorem	iunhoiioo 46674*	A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
Theorem	ioovonmbl 46675*	Any n-dimensional open interval is Lebesgue measurable. This is the first statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ*)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ*)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	iccvonmbllem 46676*	Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)
Theorem	iccvonmbl 46677*	Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)
Theorem	vonioolem1 46678*	The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))    &   ⊢ 𝐸 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )    &   ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)    &   ⊢ 𝑍 = (ℤ≥‘𝑁)    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonioolem2 46679*	The n-dimensional Lebesgue measure of open intervals. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonioo 46680*	The n-dimensional Lebesgue measure of an open interval. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	vonicclem1 46681*	The sequence of the measures of the half-open intervals converges to the measure of their intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonicclem2 46682*	The n-dimensional Lebesgue measure of closed intervals. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonicc 46683*	The n-dimensional Lebesgue measure of a closed interval. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	snvonmbl 46684	A n-dimensional singleton is Lebesgue measurable. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))    ⇒   ⊢ (𝜑 → {𝐴} ∈ dom (voln‘𝑋))
Theorem	vonn0ioo 46685*	The n-dimensional Lebesgue measure of an open interval when the dimension of the space is nonzero. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
Theorem	vonn0icc 46686*	The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘))))
Theorem	ctvonmbl 46687	Any n-dimensional countable set is Lebesgue measurable. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    ⇒   ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋))
Theorem	vonn0ioo2 46688*	The n-dimensional Lebesgue measure of an open interval when the dimension of the space is nonzero. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴(,)𝐵)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵)))
Theorem	vonsn 46689	The n-dimensional Lebesgue measure of a singleton is zero. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)
Theorem	vonn0icc2 46690*	The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴[,]𝐵)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))
Theorem	vonct 46691	The n-dimensional Lebesgue measure of any countable set is zero. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = 0)
Theorem	vitali2 46692	There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 10448). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ dom vol ⊊ 𝒫 ℝ
21.43.19.6  Measurable functions Proofs for most of the theorems in section 121 of [Fremlin1]. Real-valued functions are considered, and measurability is defined with respect to an arbitrary sigma-algebra. When the sigma-algebra on the domain is the Lebesgue measure on the reals, then all real-valued measurable functions in the sense of df-mbf 25520 are also sigma-measurable, but the definition in this section considers as measurable functions, some that are not measurable in the sense of df-mbf 25520 (see mbfpsssmf 46781 and smfmbfcex 46758).
Syntax	csmblfn 46693	Extend class notation with the class of real-valued measurable functions w.r.t. sigma-algebras.
class SMblFn
Definition	df-smblfn 46694*	Define a real-valued measurable function w.r.t. a given sigma-algebra. See Definition 121C of [Fremlin1] p. 36 and Definition 135E (b) of [Fremlin1] p. 80 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
Theorem	pimltmnf2f 46695	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
Theorem	pimltmnf2 46696*	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
Theorem	preimagelt 46697*	The preimage of a right-open, unbounded below interval, is the complement of a left-closed unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
Theorem	preimalegt 46698*	The preimage of a left-open, unbounded above interval, is the complement of a right-closed unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})
Theorem	pimconstlt0 46699*	Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound less than or equal to the constant, is the empty set. Second part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅)
Theorem	pimconstlt1 46700*	Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴)