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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmovn 39401 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 39402* 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 39403 Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))

Theoremhoi2toco 39404* 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 39405* 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 39406* 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 39407* 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 39408* 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 39409 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 39410 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 39411 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 39412* 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 39413* 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 39414 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 39415* 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 39416* 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 39417* 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 39418 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 39419* 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 39420* 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 39421* 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 39422* 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 39423* 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 39424* 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 39425* 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 39426* 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 39427* 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 39428* 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 39429* 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 39430* 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 39431 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 39432 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 39433 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 39434 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 39435* 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 39436 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 39437 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 39438 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 39439 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 39440* 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 39441* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. See ovolval 22924 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 39442* (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 39443 (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 39444* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 22924 and ovolval2 39441 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 39445 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 39446* |- ( ( 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 39447* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 39444, 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 39448* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 39444, 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 39449* |- ( 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 39450* |- ( ( 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 39451* 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 39452* 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 39453* 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 39454* 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 39455* 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 39456 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 39457* 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 39458 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 39459 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 39460* 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 39461* 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 39462* 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)

Theoremvoncl 39463 The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴𝑆)       (𝜑 → ((voln‘𝑋)‘𝐴) ∈ (0[,]+∞))

Theoremvonhoi 39464* 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.)
(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))

Theoremvonxrcl 39465 The Lebesgue measure of a set is an extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴𝑆)       (𝜑 → ((voln‘𝑋)‘𝐴) ∈ ℝ*)

Theoremvonval2 39466 Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ dom (voln‘𝑋))       (𝜑 → ((voln‘𝑋)‘𝐴) = ((voln*‘𝑋)‘𝐴))

Theoremioosshoi 39467 A n-dimensional open interval is a subset of the half-open interval with the same bounds. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
X𝑘𝑋 (𝐴(,)𝐵) ⊆ X𝑘𝑋 (𝐴[,)𝐵)

Theoremvonn0hoi 39468* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremvon0val 39469 The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ dom (voln‘∅))       (𝜑 → ((voln‘∅)‘𝐴) = 0)

Theoremvonhoire 39470* The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 (𝐴[,)𝐵)) ∈ ℝ)

Theoremiinhoiicclem 39471* 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.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   (𝜑𝐹 𝑛 ∈ ℕ X𝑘𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))))       (𝜑𝐹X𝑘𝑋 (𝐴[,]𝐵))

Theoremiinhoiicc 39472* A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ X𝑘𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))) = X𝑘𝑋 (𝐴[,]𝐵))

Theoremiunhoiioolem 39473* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)    &   (𝜑𝐹X𝑘𝑋 (𝐴(,)𝐵))    &   𝐶 = inf(ran (𝑘𝑋 ↦ ((𝐹𝑘) − 𝐴)), ℝ, < )       (𝜑𝐹 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Theoremiunhoiioo 39474* A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)       (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) = X𝑘𝑋 (𝐴(,)𝐵))

Theoremioovonmbl 39475* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ 𝑆)

Theoremiccvonmbllem 39476* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑖𝑋 ↦ ((𝐴𝑖) − (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑖𝑋 ↦ ((𝐵𝑖) + (1 / 𝑛))))       (𝜑X𝑖𝑋 ((𝐴𝑖)[,](𝐵𝑖)) ∈ 𝑆)

Theoremiccvonmbl 39477* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)[,](𝐵𝑖)) ∈ 𝑆)

Theoremvonioolem1 39478* The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))    &   𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))    &   𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑛)‘𝑘)))    &   𝐸 = inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < )    &   𝑁 = ((⌊‘(1 / 𝐸)) + 1)    &   𝑍 = (ℤ𝑁)       (𝜑𝑆 ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonioolem2 39479* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonioo 39480* 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.)
(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))

Theoremvonicclem1 39481* The sequence of the measures of the half-open intervals converges to the measure of their intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))    &   𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))       (𝜑𝑆 ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonicclem2 39482* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonicc 39483* 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.)
(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))

Theoremsnvonmbl 39484 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑 → {𝐴} ∈ dom (voln‘𝑋))

Theoremvonn0ioo 39485* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremvonn0icc 39486* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,](𝐵𝑘))))

Theoremctvonmbl 39487 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐴 ≼ ω)       (𝜑𝐴 ∈ dom (voln‘𝑋))

Theoremvonn0ioo2 39488* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   𝐼 = X𝑘𝑋 (𝐴(,)𝐵)       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘(𝐴(,)𝐵)))

Theoremvonsn 39489 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)

Theoremvonn0icc2 39490* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   𝐼 = X𝑘𝑋 (𝐴[,]𝐵)       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘(𝐴[,]𝐵)))

Theoremvonct 39491 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐴 ≼ ω)       (𝜑 → ((voln‘𝑋)‘𝐴) = 0)

Theoremvitali2 39492 There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 9076). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
dom vol ⊊ 𝒫 ℝ

20.31.19.6  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 w.r.t. df-mbf 23069 are also sigma-measurable, but the definition in this section considers as measurable functions, some that are not measurable w.r.t. df-mbf 23069 (see mbfpsssmf 39576 and smfmbfcex 39553).

Syntaxcsmblfn 39493 Extend class notation with the class of measurable functions w.r.t. sigma-algebras.
class SMblFn

Definitiondf-smblfn 39494* Define a 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)) ∈ (𝑠t dom 𝑓)})

Theorempimltmnf2 39495* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < -∞} = ∅)

Theorempreimagelt 39496* The preimage of a right-open, unbounded below interval, is the complement of a left-close, unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})

Theorempreimalegt 39497* The preimage of a left-open, unbounded above interval, is the complement of a right-close, unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})

Theorempimconstlt0 39498* Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound smaller 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.)
𝑥𝜑    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶𝐵)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)

Theorempimconstlt1 39499* 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.)
𝑥𝜑    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)

Theorempimltpnf 39500* Given a real valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)

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