P. 260 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nulmbl2 25901*	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(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Theorem	unmbl 25902	A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
Theorem	shftmbl 25903*	A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ) → {𝑥 ∈ ℝ ∣ (𝑥 − 𝐵) ∈ 𝐴} ∈ dom vol)
Theorem	0mbl 25904	The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ∅ ∈ dom vol
Theorem	rembl 25905	The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ℝ ∈ dom vol
Theorem	unidmvol 25906	The union of the Lebesgue measurable sets is ℝ. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ ∪ dom vol = ℝ
Theorem	inmbl 25907	An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∩ 𝐵) ∈ dom vol)
Theorem	difmbl 25908	A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∖ 𝐵) ∈ dom vol)
Theorem	finiunmbl 25909*	A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ dom vol) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol)
Theorem	volun 25910	The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))
Theorem	volinun 25911	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‘(𝐴 ∩ 𝐵)) + (vol‘(𝐴 ∪ 𝐵))))
Theorem	volfiniun 25912*	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 ∧ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))
Theorem	iundisj 25913*	Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	iundisj2 25914*	A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	voliunlem1 25915*	Lemma for voliun 25919. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶dom vol)    &   ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛))))    &   ⊢ (𝜑 → 𝐸 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝐸))
Theorem	voliunlem2 25916*	Lemma for voliun 25919. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶dom vol)    &   ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))    ⇒   ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)
Theorem	voliunlem3 25917*	Lemma for voliun 25919. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶dom vol)    &   ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))    &   ⊢ 𝑆 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))    &   ⊢ (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Theorem	iunmbl 25918	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)
Theorem	voliun 25919	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*, < ))
Theorem	volsuplem 25920*	Lemma for volsup 25921. (Contributed by Mario Carneiro, 4-Jul-2014.)
⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴))) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))
Theorem	volsup 25921*	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 (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (vol‘∪ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
Theorem	iunmbl2 25922*	The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) → ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
Theorem	ioombl1lem1 25923*	Lemma for ioombl1 25927. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ 𝐵 = (𝐴(,)+∞)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))    &   ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)    ⇒   ⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
Theorem	ioombl1lem2 25924*	Lemma for ioombl1 25927. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ 𝐵 = (𝐴(,)+∞)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))    &   ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)    ⇒   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
Theorem	ioombl1lem3 25925*	Lemma for ioombl1 25927. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ 𝐵 = (𝐴(,)+∞)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))    &   ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)    ⇒   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
Theorem	ioombl1lem4 25926*	Lemma for ioombl1 25927. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ 𝐵 = (𝐴(,)+∞)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))    &   ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)    ⇒   ⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
Theorem	ioombl1 25927	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)
Theorem	icombl1 25928	A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
Theorem	icombl 25929	A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ∈ dom vol)
Theorem	ioombl 25930	An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴(,)𝐵) ∈ dom vol
Theorem	iccmbl 25931	A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
Theorem	iccvolcl 25932	A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)
Theorem	ovolioo 25933	The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
Theorem	volioo 25934	The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
Theorem	ioovolcl 25935	An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
Theorem	ovolfs2 25936	Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)    ⇒   ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))
Theorem	ioorcl2 25937	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))
Theorem	ioorf 25938	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*))
Theorem	ioorval 25939*	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 (,) → (𝐹‘𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
Theorem	ioorinv2 25940*	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⟩)
Theorem	ioorinv 25941*	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 (,) → ((,)‘(𝐹‘𝐴)) = 𝐴)
Theorem	ioorcl 25942*	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) → (𝐹‘𝐴) ∈ ( ≤ ∩ (ℝ × ℝ)))
Theorem	uniiccdif 25943	A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    ⇒   ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))
Theorem	uniioovol 25944*	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 25919.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Theorem	uniiccvol 25945*	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 25919.) (Contributed by Mario Carneiro, 25-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Theorem	uniioombllem1 25946*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 25-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    ⇒   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
Theorem	uniioombllem2a 25947*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    ⇒   ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))
Theorem	uniioombllem2 25948*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by AV, 13-Sep-2020.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))    &   ⊢ 𝐾 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , (vol* ∘ 𝐻)) ⇝ (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
Theorem	uniioombllem3a 25949*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 8-May-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))    ⇒   ⊢ (𝜑 → (𝐾 = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
Theorem	uniioombllem3 25950*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))    ⇒   ⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐴)) + (vol*‘(𝐸 ∖ 𝐴))) < (((vol*‘(𝐾 ∩ 𝐴)) + (vol*‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)))
Theorem	uniioombllem4 25951*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))    &   ⊢ 𝐿 = ∪ (((,) ∘ 𝐹) “ (1...𝑁))    ⇒   ⊢ (𝜑 → (vol*‘(𝐾 ∩ 𝐴)) ≤ ((vol*‘(𝐾 ∩ 𝐿)) + 𝐶))
Theorem	uniioombllem5 25952*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))    &   ⊢ 𝐿 = ∪ (((,) ∘ 𝐹) “ (1...𝑁))    ⇒   ⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐴)) + (vol*‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Theorem	uniioombllem6 25953*	Lemma for uniioombl 25954. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)    &   ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    ⇒   ⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐴)) + (vol*‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Theorem	uniioombl 25954*	A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25918.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ∈ dom vol)
Theorem	uniiccmbl 25955*	An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25918.) (Contributed by Mario Carneiro, 25-Mar-2015.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ∈ dom vol)
Theorem	dyadf 25956*	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))
Theorem	dyadval 25957*	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))⟩)
Theorem	dyadovol 25958*	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)))
Theorem	dyadss 25959*	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 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) → 𝐷 ≤ 𝐶))
Theorem	dyaddisjlem 25960*	Lemma for dyaddisj 25961. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    ⇒   ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) ∧ 𝐶 ≤ 𝐷) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) ∨ ([,]‘(𝐵𝐹𝐷)) ⊆ ([,]‘(𝐴𝐹𝐶)) ∨ (((,)‘(𝐴𝐹𝐶)) ∩ ((,)‘(𝐵𝐹𝐷))) = ∅))
Theorem	dyaddisj 25961*	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 𝐹 ∧ 𝐵 ∈ ran 𝐹) → (([,]‘𝐴) ⊆ ([,]‘𝐵) ∨ ([,]‘𝐵) ⊆ ([,]‘𝐴) ∨ (((,)‘𝐴) ∩ ((,)‘𝐵)) = ∅))
Theorem	dyadmaxlem 25962*	Lemma for dyadmax 25963. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → ¬ 𝐷 < 𝐶)    &   ⊢ (𝜑 → ([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))
Theorem	dyadmax 25963*	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 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))
Theorem	dyadmbllem 25964*	Lemma for dyadmbl 25965. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   ⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)    ⇒   ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))
Theorem	dyadmbl 25965*	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 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)    ⇒   ⊢ (𝜑 → ∪ ([,] “ 𝐴) ∈ dom vol)
Theorem	opnmbllem 25966*	Lemma for opnmbl 25967. (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)
Theorem	opnmbl 25967	All open sets are measurable. This proof, via dyadmbl 25965 and uniioombl 25954, 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)
Theorem	opnmblALT 25968	All open sets are measurable. This alternative proof of opnmbl 25967 is significantly shorter, at the expense of invoking countable choice ax-cc 10550. (This was also the original proof before the current opnmbl 25967 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)
Theorem	subopnmbl 25969	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 ∧ 𝐵 ∈ 𝐽) → 𝐵 ∈ dom vol)
Theorem	volsup2 25970*	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))))
Theorem	volcn 25971*	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))
Theorem	volivth 25972*	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(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
Theorem	vitalilem1 25973*	Lemma for vitali 25978. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    ⇒   ⊢ ∼ Er (0[,]1)
Theorem	vitalilem2 25974*	Lemma for vitali 25978. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    &   ⊢ 𝑆 = ((0[,]1) / ∼ )    &   ⊢ (𝜑 → 𝐹 Fn 𝑆)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))    &   ⊢ (𝜑 → 𝐺:ℕ⟶1-1onto(ℚ ∩ ( -1[,]1)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})    &   ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))    ⇒   ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ( -1[,]2)))
Theorem	vitalilem3 25975*	Lemma for vitali 25978. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    &   ⊢ 𝑆 = ((0[,]1) / ∼ )    &   ⊢ (𝜑 → 𝐹 Fn 𝑆)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))    &   ⊢ (𝜑 → 𝐺:ℕ⟶1-1onto(ℚ ∩ ( -1[,]1)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})    &   ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))    ⇒   ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
Theorem	vitalilem4 25976*	Lemma for vitali 25978. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    &   ⊢ 𝑆 = ((0[,]1) / ∼ )    &   ⊢ (𝜑 → 𝐹 Fn 𝑆)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))    &   ⊢ (𝜑 → 𝐺:ℕ⟶1-1onto(ℚ ∩ ( -1[,]1)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})    &   ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))    ⇒   ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)
Theorem	vitalilem5 25977*	Lemma for vitali 25978. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    &   ⊢ 𝑆 = ((0[,]1) / ∼ )    &   ⊢ (𝜑 → 𝐹 Fn 𝑆)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))    &   ⊢ (𝜑 → 𝐺:ℕ⟶1-1onto(ℚ ∩ ( -1[,]1)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})    &   ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))    ⇒   ⊢ ¬ 𝜑
Theorem	vitali 25978	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
Syntax	cmbf 25979	Extend class notation with the class of measurable functions.
class MblFn
Syntax	citg1 25980	Extend class notation with the Lebesgue integral for simple functions.
class ∫1
Syntax	citg2 25981	Extend class notation with the Lebesgue integral for nonnegative functions.
class ∫2
Syntax	cibl 25982	Extend class notation with the class of integrable functions.
class 𝐿1
Syntax	citg 25983	Extend class notation with the general Lebesgue integral.
class ∫𝐴𝐵 d𝑥
Definition	df-mbf 25984*	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 25891) 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)}
Definition	df-itg1 25985*	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}))))
Definition	df-itg2 25986*	Define the Lebesgue integral for nonnegative functions. A nonnegative function's integral is the supremum of the integrals of all simple functions that are less than the input function. Note that this may be +∞ for functions that take the value +∞ on a set of positive measure or functions that are bounded below by a positive number on a set of infinite measure. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ∫2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
Definition	df-ibl 25987*	Define the class of integrable functions on the reals. A function is integrable if it is measurable and the integrals of the pieces of the function are all finite. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
Definition	df-itg 25988*	Define the full Lebesgue integral, for complex-valued functions to ℝ. The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of 𝑥↑2 from 0 to 1 is ∫(0[,]1)(𝑥↑2) d𝑥 = (1 / 3). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 25986 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 25986 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
Theorem	ismbf1 25989*	The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25993 and ismbfcn 25994 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
Theorem	mbff 25990	A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
Theorem	mbfdm 25991	The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
Theorem	mbfconstlem 25992	Lemma for mbfconst 25998 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol)
Theorem	ismbf 25993*	The predicate "𝐹 is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 25891. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
Theorem	ismbfcn 25994	A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
Theorem	mbfima 25995	Definitional property of a measurable function: the preimage of an open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ (𝐵(,)𝐶)) ∈ dom vol)
Theorem	mbfimaicc 25996	The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (𝐵[,]𝐶)) ∈ dom vol)
Theorem	mbfimasn 25997	The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ ∧ 𝐵 ∈ ℝ) → (◡𝐹 “ {𝐵}) ∈ dom vol)
Theorem	mbfconst 25998	A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn)
Theorem	mbf0 25999	The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
⊢ ∅ ∈ MblFn
Theorem	mbfid 26000	The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn)