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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovolfsf 24001 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
 
Theoremovolsf 24002 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)    &   𝑆 = seq1( + , 𝐺)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
 
Theoremovolval 24003* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 
Theoremelovolmlem 24004 Lemma for elovolm 24005 and related theorems. (Contributed by BJ, 23-Jul-2022.)
(𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ)))
 
Theoremelovolm 24005* Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
 
Theoremelovolmr 24006* Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolmge0 24007* The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝐵𝑀 → 0 ≤ 𝐵)
 
Theoremovolcl 24008 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
 
Theoremovollb 24009 The outer volume is a lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolgelb 24010* The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))       ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
 
Theoremovolge0 24011 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
 
Theoremovolf 24012 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
vol*:𝒫 ℝ⟶(0[,]+∞)
 
Theoremovollecl 24013 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (vol*‘𝐴) ≤ 𝐵) → (vol*‘𝐴) ∈ ℝ)
 
Theoremovolsslem 24014* Lemma for ovolss 24015. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolss 24015 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolsscl 24016 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ)
 
Theoremovolssnul 24017 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)
 
Theoremovollb2lem 24018* Lemma for ovollb2 24019. (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 24019 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 24009). (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 24020 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 24021 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘ℚ) = 0
 
Theoremovolctb2 24022 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)
 
Theoremovol0 24023 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘∅) = 0
 
Theoremovolfi 24024 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)
 
Theoremovolsn 24025 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
 
Theoremovolunlem1a 24026* Lemma for ovolun 24029. (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 24027* Lemma for ovolun 24029. (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 24028 Lemma for ovolun 24029. (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 24029 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 24035, 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 24030 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 24031* 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 24032* Lemma for ovoliun 24035. (Contributed by Mario Carneiro, 12-Jun-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 24033* Lemma for ovoliun 24035. (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 24034* Lemma for ovoliun 24035. (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 24035* 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 24015, 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 24036* 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 24035.) (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 24037* 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 24038* 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 24039* Lemma for ovolshft 24041. (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 24040* Lemma for ovolshft 24041. (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 24041* 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 24042* 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 24043* Lemma for ovolsca 24045. (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 24044* Lemma for ovolshft 24041. (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 24045* 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 24046* 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 24047* Lemma for ovolicc2 24052. (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 24048* Lemma for ovolicc2 24052. (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 24049* Lemma for ovolicc2 24052. (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 24050* Lemma for ovolicc2 24052. (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 24051* Lemma for ovolicc2 24052. (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 24052* 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 24053 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵𝐴))
 
Theoremovolicopnf 24054 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)
 
Theoremovolre 24055 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
(vol*‘ℝ) = +∞
 
Theoremismbl 24056* 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 24057* From ovolun 24029, 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 24058 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol = (vol* ↾ dom vol)
 
Theoremvolf 24059 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol:dom vol⟶(0[,]+∞)
 
Theoremmblvol 24060 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 24061 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
 
Theoremmblsplit 24062 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 24063 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 24064 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)
 
Theoremnulmbl 24065 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)
 
Theoremnulmbl2 24066* 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 24067 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremshftmbl 24068* 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 24069 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
∅ ∈ dom vol
 
Theoremrembl 24070 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
ℝ ∈ dom vol
 
Theoremunidmvol 24071 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)
dom vol = ℝ
 
Theoreminmbl 24072 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremdifmbl 24073 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremfiniunmbl 24074* 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 24075 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 24076 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 24077* 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 24078* 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 24079* 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 24080* Lemma for voliun 24084. (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 24081* Lemma for voliun 24084. (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 24082* Lemma for voliun 24084. (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 24083 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 24084 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 24085* Lemma for volsup 24086. (Contributed by Mario Carneiro, 4-Jul-2014.)
((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴))) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremvolsup 24086* 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 24087* 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 24088* Lemma for ioombl1 24092. (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 24089* Lemma for ioombl1 24092. (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 24090* Lemma for ioombl1 24092. (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 24091* Lemma for ioombl1 24092. (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 24092 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 24093 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
 
Theoremicombl 24094 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 24095 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴(,)𝐵) ∈ dom vol
 
Theoremiccmbl 24096 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
 
Theoremiccvolcl 24097 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)
 
Theoremovolioo 24098 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵𝐴))
 
Theoremvolioo 24099 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵𝐴))
 
Theoremioovolcl 24100 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
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