P. 232 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ismbl 23101*	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 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 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*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
Theorem	ismbl2 23102*	From ovolun 23074, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
Theorem	volres 23103	A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ vol = (vol* ↾ dom vol)
Theorem	volf 23104	The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ vol:dom vol⟶(0[,]+∞)
Theorem	mblvol 23105	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))
Theorem	mblss 23106	A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
Theorem	mblsplit 23107	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 23108	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 23109	The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ (𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)
Theorem	nulmbl 23110	A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)
Theorem	nulmbl2 23111*	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 23112	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 23113*	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 23114	The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ∅ ∈ dom vol
Theorem	rembl 23115	The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ℝ ∈ dom vol
Theorem	unidmvol 23116	The union of the Lebesgue measurable sets is ℝ. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ ∪ dom vol = ℝ
Theorem	inmbl 23117	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 23118	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 23119*	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 23120	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 23121	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 23122*	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 23123*	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 23124*	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 23125*	Lemma for voliun 23129. (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 23126*	Lemma for voliun 23129. (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 23127*	Lemma for voliun 23129. (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 23128	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 23129	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 23130*	Lemma for volsup 23131. (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 23131*	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 23132*	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 23133*	Lemma for ioombl1 23137. (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 23134*	Lemma for ioombl1 23137. (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 23135*	Lemma for ioombl1 23137. (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 23136*	Lemma for ioombl1 23137. (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 23137	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 23138	A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
Theorem	icombl 23139	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 23140	An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ (𝐴(,)𝐵) ∈ dom vol
Theorem	iccmbl 23141	A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
Theorem	iccvolcl 23142	A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ)
Theorem	ovolioo 23143	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	ioovolcl 23144	An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
Theorem	ovolfs2 23145	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 23146	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 23147	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 23148*	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 23149*	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 23150*	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 23151*	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 23152	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 23153*	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 23129.) 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 23154*	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 23129.) (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 23155*	Lemma for uniioombl 23163. (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 23156*	Lemma for uniioombl 23163. (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 23157*	Lemma for uniioombl 23163. (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 23158*	Lemma for uniioombl 23163. (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 23159*	Lemma for uniioombl 23163. (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 23160*	Lemma for uniioombl 23163. (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 23161*	Lemma for uniioombl 23163. (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 23162*	Lemma for uniioombl 23163. (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 23163*	A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23128.) 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 23164*	An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23128.) (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 23165*	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 23166*	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 23167*	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 23168*	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 23169*	Lemma for dyaddisj 23170. (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 23170*	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 23171*	Lemma for dyadmax 23172. (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 23172*	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 23173*	Lemma for dyadmbl 23174. (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 23174*	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 23175*	Lemma for opnmbl 23176. (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 23176	All open sets are measurable. This proof, via dyadmbl 23174 and uniioombl 23163, 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 23177	All open sets are measurable. This alternative proof of opnmbl 23176 is significantly shorter, at the expense of invoking countable choice ax-cc 9140. (This was also the original proof before the current opnmbl 23176 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 23178	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 23179*	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 23180*	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 23181*	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 23182*	Lemma for vitali 23188. (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	vitalilem1OLD 23183*	Obsolete proof of vitalilem1 23182 as of 1-May-2021. Lemma for vitali 23188. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}    ⇒   ⊢ ∼ Er (0[,]1)
Theorem	vitalilem2 23184*	Lemma for vitali 23188. (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 23185*	Lemma for vitali 23188. (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 23186*	Lemma for vitali 23188. (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 23187*	Lemma for vitali 23188. (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 23188	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 23189	Extend class notation with the class of measurable functions.
class MblFn
Syntax	citg1 23190	Extend class notation with the Lebesgue integral for simple functions.
class ∫1
Syntax	citg2 23191	Extend class notation with the Lebesgue integral for nonnegative functions.
class ∫2
Syntax	cibl 23192	Extend class notation with the class of integrable functions.
class 𝐿1
Syntax	citg 23193	Extend class notation with the general Lebesgue integral.
class ∫𝐴𝐵 d𝑥
Definition	df-mbf 23194*	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 23101) 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 23195*	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 23196*	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 23197*	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 23198*	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 23196 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 23196 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 23199*	The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 23203 and ismbfcn 23204 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 23200	A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)