HomeHome Metamath Proof Explorer
Theorem List (p. 396 of 424)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27159)
  Hilbert Space Explorer  Hilbert Space Explorer
(27160-28684)
  Users' Mathboxes  Users' Mathboxes
(28685-42360)
 

Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovncvr2 39501* 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 39502 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 39503 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 39504 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 39505* 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 39506* 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 39507 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 39508* 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 39509* 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 39510* 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 39511 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 39512* 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 39513* 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 39514* 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 39515* 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 39516* 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 39517* 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 39518* 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 39519* 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 39520* 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 39521* 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 39522* 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 39523* 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 39524 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 39525 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 39526 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 39527 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 39528* 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 39529 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 39530 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 39531 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 39532 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 39533* 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 39534* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. See ovolval 23049 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 39535* (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 39536 (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 39537* The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23049 and ovolval2 39534 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 39538 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 39539* |- ( ( 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 39540* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 39537, 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 39541* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 39537, 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 39542* |- ( 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 39543* |- ( ( 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 39544* 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 39545* 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 39546* 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 39547* 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 39548* 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 39549 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 39550* 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 39551 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 39552 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 39553* 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 39554* 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 39555* 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 39556 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 39557* 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 39558 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 39559 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 39560 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 39561* 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 39562 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 39563* 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 39564* 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 39565* 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 39566* 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 39567* 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 39568* 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 39569* 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 39570* 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 39571* 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 39572* 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 39573* 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 39574* 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 39575* 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 39576* 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 39577 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 39578* 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 39579* 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 39580 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 39581* 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 39582 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 39583* 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 39584 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 39585 There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 9200). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
dom vol ⊊ 𝒫 ℝ
 
21.31.19.6  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 23194 are also sigma-measurable, but the definition in this section considers as measurable functions, some that are not measurable w.r.t. df-mbf 23194 (see mbfpsssmf 39669 and smfmbfcex 39646).

 
Syntaxcsmblfn 39586 Extend class notation with the class of measurable functions w.r.t. sigma-algebras.
class SMblFn
 
Definitiondf-smblfn 39587* 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 39588* 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 39589* 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 39590* 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 39591* 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 39592* 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 39593* 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)
 
Theorempimgtpnf2 39594* Given a real valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
 
Theoremsalpreimagelt 39595* If all the preimages of left-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
 
Theorempimrecltpos 39596 The preimage of an unbounded below, open interval, with positive upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → {𝑥𝐴 ∣ (1 / 𝐵) < 𝐶} = ({𝑥𝐴 ∣ (1 / 𝐶) < 𝐵} ∪ {𝑥𝐴𝐵 < 0}))
 
Theoremsalpreimalegt 39597* If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)
 
Theorempimiooltgt 39598* The preimage of an open interval is the intersection of the preimage of an unbounded below open interval and an unbounded above open interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥𝐴𝐵 < 𝑅} ∩ {𝑥𝐴𝐿 < 𝐵}))
 
Theorempreimaicomnf 39599* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞[,)𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) < 𝐵})
 
Theorempimltpnf2 39600* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
  Copyright terms: Public domain < Previous  Next >