P. 247 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mblsplit 24601	The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴))))
Theorem	volss 24602	The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵) → (vol‘𝐴) ≤ (vol‘𝐵))
Theorem	cmmbl 24603	The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ (𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)
Theorem	nulmbl 24604	A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)
Theorem	nulmbl2 24605*	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 24606	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 24607*	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 24608	The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ∅ ∈ dom vol
Theorem	rembl 24609	The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ℝ ∈ dom vol
Theorem	unidmvol 24610	The union of the Lebesgue measurable sets is ℝ. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ ∪ dom vol = ℝ
Theorem	inmbl 24611	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 24612	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 24613*	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 24614	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 24615	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 24616*	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 24617*	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 24618*	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 24619*	Lemma for voliun 24623. (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 24620*	Lemma for voliun 24623. (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 24621*	Lemma for voliun 24623. (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 24622	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 24623	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 24624*	Lemma for volsup 24625. (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 24625*	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 24626*	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 24627*	Lemma for ioombl1 24631. (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 24628*	Lemma for ioombl1 24631. (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 24629*	Lemma for ioombl1 24631. (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 24630*	Lemma for ioombl1 24631. (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 24631	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 24632	A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
Theorem	icombl 24633	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 24634	An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴(,)𝐵) ∈ dom vol
Theorem	iccmbl 24635	A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
Theorem	iccvolcl 24636	A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)
Theorem	ovolioo 24637	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 24638	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 24639	An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
Theorem	ovolfs2 24640	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 24641	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 24642	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 24643*	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 24644*	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 24645*	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 24646*	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 24647	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 24648*	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 24623.) 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 24649*	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 24623.) (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 24650*	Lemma for uniioombl 24658. (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 24651*	Lemma for uniioombl 24658. (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 24652*	Lemma for uniioombl 24658. (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 24653*	Lemma for uniioombl 24658. (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 24654*	Lemma for uniioombl 24658. (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 24655*	Lemma for uniioombl 24658. (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 24656*	Lemma for uniioombl 24658. (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 24657*	Lemma for uniioombl 24658. (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 24658*	A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24622.) 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 24659*	An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24622.) (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 24660*	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 24661*	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 24662*	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 24663*	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 24664*	Lemma for dyaddisj 24665. (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 24665*	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 24666*	Lemma for dyadmax 24667. (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 24667*	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 24668*	Lemma for dyadmbl 24669. (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 24669*	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 24670*	Lemma for opnmbl 24671. (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 24671	All open sets are measurable. This proof, via dyadmbl 24669 and uniioombl 24658, 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 24672	All open sets are measurable. This alternative proof of opnmbl 24671 is significantly shorter, at the expense of invoking countable choice ax-cc 10122. (This was also the original proof before the current opnmbl 24671 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 24673	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 24674*	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 24675*	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 24676*	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 24677*	Lemma for vitali 24682. (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 24678*	Lemma for vitali 24682. (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–1-1-onto→(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 24679*	Lemma for vitali 24682. (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–1-1-onto→(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 24680*	Lemma for vitali 24682. (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–1-1-onto→(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 24681*	Lemma for vitali 24682. (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–1-1-onto→(ℚ ∩ (-1[,]1)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})    &   ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))    ⇒   ⊢ ¬ 𝜑
Theorem	vitali 24682	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 24683	Extend class notation with the class of measurable functions.
class MblFn
Syntax	citg1 24684	Extend class notation with the Lebesgue integral for simple functions.
class ∫1
Syntax	citg2 24685	Extend class notation with the Lebesgue integral for nonnegative functions.
class ∫2
Syntax	cibl 24686	Extend class notation with the class of integrable functions.
class 𝐿1
Syntax	citg 24687	Extend class notation with the general Lebesgue integral.
class ∫𝐴𝐵 d𝑥
Definition	df-mbf 24688*	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 24595) 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 24689*	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 24690*	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 24691*	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 24692*	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 24690 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 24690 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 24693*	The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24697 and ismbfcn 24698 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 24694	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 24695	The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
Theorem	mbfconstlem 24696	Lemma for mbfconst 24702 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol)
Theorem	ismbf 24697*	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 24595. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
Theorem	ismbfcn 24698	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 24699	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 24700	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)