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Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovolssnul 23301 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)
 
Theoremovollb2lem 23302* Lemma for ovollb2 23303. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ([,] ∘ 𝐹))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)       (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
 
Theoremovollb2 23303 It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 23293). (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolctb 23304 The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
 
Theoremovolq 23305 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘ℚ) = 0
 
Theoremovolctb2 23306 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)
 
Theoremovol0 23307 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘∅) = 0
 
Theoremovolfi 23308 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)
 
Theoremovolsn 23309 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
 
Theoremovolunlem1a 23310* Lemma for ovolun 23313. (Contributed by Mario Carneiro, 7-May-2015.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   (𝜑𝐵 ran ((,) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))    &   𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))       ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolunlem1 23311* Lemma for ovolun 23313. (Contributed by Mario Carneiro, 12-Jun-2014.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   (𝜑𝐵 ran ((,) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))    &   𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))       (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolunlem2 23312 Lemma for ovolun 23313. (Contributed by Mario Carneiro, 12-Jun-2014.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolun 23313 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 23319, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
(((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
 
Theoremovolunnul 23314 Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) = (vol*‘𝐴))
 
Theoremovolfiniun 23315* The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵))
 
Theoremovoliunlem1 23316* Lemma for ovoliun 23319. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))    &   (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))    &   ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿)       (𝜑 → (𝑈𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliunlem2 23317* Lemma for ovoliun 23319. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))    &   (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))    &   ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliunlem3 23318* Lemma for ovoliun 23319. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliun 23319* The Lebesgue outer measure function is countably sub-additive. (Many books allow +∞ as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 23299, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
 
Theoremovoliun2 23320* The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 23319.) (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑𝑇 ∈ dom ⇝ )       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
 
Theoremovoliunnul 23321* A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
 
Theoremshft2rab 23322* If 𝐵 is a shift of 𝐴 by 𝐶, then 𝐴 is a shift of 𝐵 by -𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})       (𝜑𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵})
 
Theoremovolshftlem1 23323* Lemma for ovolshft 23325. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))       (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolshftlem2 23324* Lemma for ovolshft 23325. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀)
 
Theoremovolshft 23325* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})       (𝜑 → (vol*‘𝐴) = (vol*‘𝐵))
 
Theoremsca2rab 23326* If 𝐵 is a scale of 𝐴 by 𝐶, then 𝐴 is a scale of 𝐵 by 1 / 𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})       (𝜑𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵})
 
Theoremovolscalem1 23327* Lemma for ovolsca 23329. (Contributed by Mario Carneiro, 6-Apr-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))       (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
 
Theoremovolscalem2 23328* Lemma for ovolshft 23325. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶))
 
Theoremovolsca 23329* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶))
 
Theoremovolicc1 23330* The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))       (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
 
Theoremovolicc2lem1 23331* Lemma for ovolicc2 23336. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)       ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
 
Theoremovolicc2lem2 23332* Lemma for ovolicc2 23336. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}       ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵)
 
Theoremovolicc2lem3 23333* Lemma for ovolicc2 23336. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}       ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃))))))
 
Theoremovolicc2lem4 23334* Lemma for ovolicc2 23336. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}    &   𝑀 = inf(𝑊, ℝ, < )       (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolicc2lem5 23335* Lemma for ovolicc2 23336. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}       (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolicc2 23336* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝜑 → (𝐵𝐴) ≤ (vol*‘(𝐴[,]𝐵)))
 
Theoremovolicc 23337 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵𝐴))
 
Theoremovolicopnf 23338 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)
 
Theoremovolre 23339 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
(vol*‘ℝ) = +∞
 
Theoremismbl 23340* The predicate "𝐴 is Lebesgue-measurable". 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 (assuming that the measure of 𝑥 is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
 
Theoremismbl2 23341* From ovolun 23313, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥𝐴 and 𝑥𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
 
Theoremvolres 23342 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol = (vol* ↾ dom vol)
 
Theoremvolf 23343 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol:dom vol⟶(0[,]+∞)
 
Theoremmblvol 23344 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
 
Theoremmblss 23345 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
 
Theoremmblsplit 23346 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) = ((vol*‘(𝐵𝐴)) + (vol*‘(𝐵𝐴))))
 
Theoremvolss 23347 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴𝐵) → (vol‘𝐴) ≤ (vol‘𝐵))
 
Theoremcmmbl 23348 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)
 
Theoremnulmbl 23349 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)
 
Theoremnulmbl2 23350* A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has vol*(𝐴) = 0 while "outer measure zero" means that for any 𝑥 there is a 𝑦 containing 𝐴 with volume less than 𝑥. Assuming AC, these notions are equivalent (because the intersection of all such 𝑦 is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)
(∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
 
Theoremunmbl 23351 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremshftmbl 23352* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ) → {𝑥 ∈ ℝ ∣ (𝑥𝐵) ∈ 𝐴} ∈ dom vol)
 
Theorem0mbl 23353 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
∅ ∈ dom vol
 
Theoremrembl 23354 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
ℝ ∈ dom vol
 
Theoremunidmvol 23355 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)
dom vol = ℝ
 
Theoreminmbl 23356 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremdifmbl 23357 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremfiniunmbl 23358* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ dom vol) → 𝑘𝐴 𝐵 ∈ dom vol)
 
Theoremvolun 23359 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
(((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))
 
Theoremvolinun 23360 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)
(((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol‘(𝐴𝐵)) + (vol‘(𝐴𝐵))))
 
Theoremvolfiniun 23361* The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))
 
Theoremiundisj 23362* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremiundisj2 23363* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
(𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremvoliunlem1 23364* Lemma for voliun 23368. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝜑𝐹:ℕ⟶dom vol)    &   (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))    &   𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)       ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
 
Theoremvoliunlem2 23365* Lemma for voliun 23368. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝜑𝐹:ℕ⟶dom vol)    &   (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))    &   𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))       (𝜑 ran 𝐹 ∈ dom vol)
 
Theoremvoliunlem3 23366* Lemma for voliun 23368. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝜑𝐹:ℕ⟶dom vol)    &   (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))    &   𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))    &   𝑆 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛)))    &   (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ)       (𝜑 → (vol‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
 
Theoremiunmbl 23367 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
 
Theoremvoliun 23368 The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑆 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))       ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))
 
Theoremvolsuplem 23369* Lemma for volsup 23370. (Contributed by Mario Carneiro, 4-Jul-2014.)
((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴))) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremvolsup 23370* The volume of the limit of an increasing sequence of measurable sets is the limit of the volumes. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
 
Theoremiunmbl2 23371* The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 𝐵 ∈ dom vol) → 𝑛𝐴 𝐵 ∈ dom vol)
 
Theoremioombl1lem1 23372* Lemma for ioombl1 23376. (Contributed by Mario Carneiro, 18-Aug-2014.)
𝐵 = (𝐴(,)+∞)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   𝑃 = (1st ‘(𝐹𝑛))    &   𝑄 = (2nd ‘(𝐹𝑛))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)       (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
 
Theoremioombl1lem2 23373* Lemma for ioombl1 23376. (Contributed by Mario Carneiro, 18-Aug-2014.)
𝐵 = (𝐴(,)+∞)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   𝑃 = (1st ‘(𝐹𝑛))    &   𝑄 = (2nd ‘(𝐹𝑛))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)       (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
 
Theoremioombl1lem3 23374* Lemma for ioombl1 23376. (Contributed by Mario Carneiro, 18-Aug-2014.)
𝐵 = (𝐴(,)+∞)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   𝑃 = (1st ‘(𝐹𝑛))    &   𝑄 = (2nd ‘(𝐹𝑛))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)       ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
 
Theoremioombl1lem4 23375* Lemma for ioombl1 23376. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝐵 = (𝐴(,)+∞)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   𝑃 = (1st ‘(𝐹𝑛))    &   𝑄 = (2nd ‘(𝐹𝑛))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)       (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
 
Theoremioombl1 23376 An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
(𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)
 
Theoremicombl1 23377 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
 
Theoremicombl 23378 A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ∈ dom vol)
 
Theoremioombl 23379 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴(,)𝐵) ∈ dom vol
 
Theoremiccmbl 23380 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
 
Theoremiccvolcl 23381 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)
 
Theoremovolioo 23382 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵𝐴))
 
Theoremvolioo 23383 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵𝐴))
 
Theoremioovolcl 23384 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
 
Theoremovolfs2 23385 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))
 
Theoremioorcl2 23386 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
(((𝐴(,)𝐵) ≠ ∅ ∧ (vol*‘(𝐴(,)𝐵)) ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
 
Theoremioorf 23387 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       𝐹:ran (,)⟶( ≤ ∩ (ℝ* × ℝ*))
 
Theoremioorval 23388* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       (𝐴 ∈ ran (,) → (𝐹𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
 
Theoremioorinv2 23389* The function 𝐹 is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = ⟨𝐴, 𝐵⟩)
 
Theoremioorinv 23390* The function 𝐹 is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       (𝐴 ∈ ran (,) → ((,)‘(𝐹𝐴)) = 𝐴)
 
Theoremioorcl 23391* The function 𝐹 does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹𝐴) ∈ ( ≤ ∩ (ℝ × ℝ)))
 
Theoremuniiccdif 23392 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))       (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
 
Theoremuniioovol 23393* A disjoint union of open intervals has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 23368.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
 
Theoremuniiccvol 23394* An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 23368.) (Contributed by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
 
Theoremuniioombllem1 23395* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))       (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
 
Theoremuniioombllem2a 23396* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 7-May-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))       (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
 
Theoremuniioombllem2 23397* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by AV, 13-Sep-2020.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))))    &   𝐾 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))       ((𝜑𝐽 ∈ ℕ) → seq1( + , (vol* ∘ 𝐻)) ⇝ (vol*‘(((,)‘(𝐺𝐽)) ∩ 𝐴)))
 
Theoremuniioombllem3a 23398* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 8-May-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))       (𝜑 → (𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
 
Theoremuniioombllem3 23399* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))       (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
 
Theoremuniioombllem4 23400* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))    &   𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))       (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
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