Home Metamath Proof ExplorerTheorem List (p. 242 of 451) < 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 (1-28677) Hilbert Space Explorer (28678-30200) Users' Mathboxes (30201-45078)

Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremovolicc2lem5 24101* Lemma for ovolicc2 24102. (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 24102* 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 24103 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵𝐴))

Theoremovolicopnf 24104 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)

Theoremovolre 24105 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
(vol*‘ℝ) = +∞

Theoremismbl 24106* 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 24107* From ovolun 24079, 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 24108 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol = (vol* ↾ dom vol)

Theoremvolf 24109 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol:dom vol⟶(0[,]+∞)

Theoremmblvol 24110 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 24111 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)

Theoremmblsplit 24112 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 24113 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 24114 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)

Theoremnulmbl 24115 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)

Theoremnulmbl2 24116* 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 24117 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremshftmbl 24118* 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 24119 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
∅ ∈ dom vol

Theoremrembl 24120 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
ℝ ∈ dom vol

Theoremunidmvol 24121 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)
dom vol = ℝ

Theoreminmbl 24122 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremdifmbl 24123 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremfiniunmbl 24124* 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 24125 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 24126 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 24127* 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 24128* 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 24129* 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 24130* Lemma for voliun 24134. (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 24131* Lemma for voliun 24134. (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 24132* Lemma for voliun 24134. (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 24133 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 24134 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 24135* Lemma for volsup 24136. (Contributed by Mario Carneiro, 4-Jul-2014.)
((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴))) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremvolsup 24136* 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 24137* 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 24138* Lemma for ioombl1 24142. (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 24139* Lemma for ioombl1 24142. (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 24140* Lemma for ioombl1 24142. (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 24141* Lemma for ioombl1 24142. (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 24142 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 24143 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)

Theoremicombl 24144 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 24145 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
(𝐴(,)𝐵) ∈ dom vol

Theoremiccmbl 24146 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)

Theoremiccvolcl 24147 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)

Theoremovolioo 24148 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵𝐴))

Theoremvolioo 24149 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵𝐴))

Theoremioovolcl 24150 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)

Theoremovolfs2 24151 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 24152 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 24153 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 24154* 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 24155* 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 24156* 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 24157* 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 24158 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 24159* 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 24134.) 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 24160* 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 24134.) (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 24161* Lemma for uniioombl 24169. (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 24162* Lemma for uniioombl 24169. (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 24163* Lemma for uniioombl 24169. (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 24164* Lemma for uniioombl 24169. (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 24165* Lemma for uniioombl 24169. (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 24166* Lemma for uniioombl 24169. (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))

Theoremuniioombllem5 24167* Lemma for uniioombl 24169. (Contributed by Mario Carneiro, 25-Aug-2014.)
(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*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Theoremuniioombllem6 24168* Lemma for uniioombl 24169. (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))       (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Theoremuniioombl 24169* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24133.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 ran ((,) ∘ 𝐹) ∈ dom vol)

Theoremuniiccmbl 24170* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24133.) (Contributed by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 ran ([,] ∘ 𝐹) ∈ dom vol)

Theoremdyadf 24171* The function 𝐹 returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))

Theoremdyadval 24172* Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)

Theoremdyadovol 24173* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (vol*‘([,]‘(𝐴𝐹𝐵))) = (1 / (2↑𝐵)))

Theoremdyadss 24174* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) → 𝐷𝐶))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) ∧ 𝐶𝐷) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) ∨ ([,]‘(𝐵𝐹𝐷)) ⊆ ([,]‘(𝐴𝐹𝐶)) ∨ (((,)‘(𝐴𝐹𝐶)) ∩ ((,)‘(𝐵𝐹𝐷))) = ∅))

Theoremdyaddisj 24176* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ran 𝐹𝐵 ∈ ran 𝐹) → (([,]‘𝐴) ⊆ ([,]‘𝐵) ∨ ([,]‘𝐵) ⊆ ([,]‘𝐴) ∨ (((,)‘𝐴) ∩ ((,)‘𝐵)) = ∅))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑 → ¬ 𝐷 < 𝐶)    &   (𝜑 → ([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Theoremdyadmax 24178* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   (𝜑𝐴 ⊆ ran 𝐹)       (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))

Theoremdyadmbl 24180* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   (𝜑𝐴 ⊆ ran 𝐹)       (𝜑 ([,] “ 𝐴) ∈ dom vol)

Theoremopnmbllem 24181* Lemma for opnmbl 24182. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Theoremopnmbl 24182 All open sets are measurable. This proof, via dyadmbl 24180 and uniioombl 24169, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

TheoremopnmblALT 24183 All open sets are measurable. This alternative proof of opnmbl 24182 is significantly shorter, at the expense of invoking countable choice ax-cc 9833. (This was also the original proof before the current opnmbl 24182 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Theoremsubopnmbl 24184 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t 𝐴)       ((𝐴 ∈ dom vol ∧ 𝐵𝐽) → 𝐵 ∈ dom vol)

Theoremvolsup2 24185* The volume of 𝐴 is the supremum of the sequence vol*‘(𝐴 ∩ (-𝑛[,]𝑛)) of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))

Theoremvolcn 24186* The function formed by restricting a measurable set to a closed interval with a varying endpoint produces an increasing continuous function on the reals. (Contributed by Mario Carneiro, 30-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ (vol‘(𝐴 ∩ (𝐵[,]𝑥))))       ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ) → 𝐹 ∈ (ℝ–cn→ℝ))

Theoremvolivth 24187* The Intermediate Value Theorem for the Lebesgue volume function. For any positive 𝐵 ≤ (vol‘𝐴), there is a measurable subset of 𝐴 whose volume is 𝐵. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))

Theoremvitalilem1 24188* Lemma for vitali 24193. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}        Er (0[,]1)

Theoremvitalilem2 24189* Lemma for vitali 24193. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))

Theoremvitalilem3 24190* Lemma for vitali 24193. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))

Theoremvitalilem4 24191* Lemma for vitali 24193. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)

Theoremvitalilem5 24192* Lemma for vitali 24193. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))        ¬ 𝜑

Theoremvitali 24193 If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)
( < We ℝ → dom vol ⊊ 𝒫 ℝ)

13.2.2  Lebesgue integration

13.2.2.1  Lesbesgue integral

Syntaxcmbf 24194 Extend class notation with the class of measurable functions.
class MblFn

Syntaxcitg1 24195 Extend class notation with the Lebesgue integral for simple functions.
class 1

Syntaxcitg2 24196 Extend class notation with the Lebesgue integral for nonnegative functions.
class 2

Syntaxcibl 24197 Extend class notation with the class of integrable functions.
class 𝐿1

Syntaxcitg 24198 Extend class notation with the general Lebesgue integral.
class 𝐴𝐵 d𝑥

Definitiondf-mbf 24199* Define the class of measurable functions on the reals. A real function is measurable if the preimage of every open interval is a measurable set (see ismbl 24106) and a complex function is measurable if the real and imaginary parts of the function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}

Definitiondf-itg1 24200* Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)
1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))))

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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45078
 Copyright terms: Public domain < Previous  Next >