P. 256 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uniioombllem3a 25501*	Lemma for uniioombl 25506. (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 25502*	Lemma for uniioombl 25506. (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 25503*	Lemma for uniioombl 25506. (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 25504*	Lemma for uniioombl 25506. (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 25505*	Lemma for uniioombl 25506. (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 25506*	A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25470.) 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 25507*	An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25470.) (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 25508*	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 25509*	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 25510*	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 25511*	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 25512*	Lemma for dyaddisj 25513. (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 25513*	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 25514*	Lemma for dyadmax 25515. (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 25515*	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 25516*	Lemma for dyadmbl 25517. (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 25517*	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 25518*	Lemma for opnmbl 25519. (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 25519	All open sets are measurable. This proof, via dyadmbl 25517 and uniioombl 25506, 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 25520	All open sets are measurable. This alternative proof of opnmbl 25519 is significantly shorter, at the expense of invoking countable choice ax-cc 10348. (This was also the original proof before the current opnmbl 25519 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 25521	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 25522*	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 25523*	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 25524*	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 25525*	Lemma for vitali 25530. (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 25526*	Lemma for vitali 25530. (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 25527*	Lemma for vitali 25530. (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 25528*	Lemma for vitali 25530. (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 25529*	Lemma for vitali 25530. (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 25530	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 25531	Extend class notation with the class of measurable functions.
class MblFn
Syntax	citg1 25532	Extend class notation with the Lebesgue integral for simple functions.
class ∫1
Syntax	citg2 25533	Extend class notation with the Lebesgue integral for nonnegative functions.
class ∫2
Syntax	cibl 25534	Extend class notation with the class of integrable functions.
class 𝐿1
Syntax	citg 25535	Extend class notation with the general Lebesgue integral.
class ∫𝐴𝐵 d𝑥
Definition	df-mbf 25536*	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 25443) 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 25537*	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 25538*	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 25539*	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 25540*	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 25538 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 25538 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 25541*	The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25545 and ismbfcn 25546 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 25542	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 25543	The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
Theorem	mbfconstlem 25544	Lemma for mbfconst 25550 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol)
Theorem	ismbf 25545*	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 25443. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
Theorem	ismbfcn 25546	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 25547	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 25548	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 25549	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 25550	A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn)
Theorem	mbf0 25551	The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
⊢ ∅ ∈ MblFn
Theorem	mbfid 25552	The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn)
Theorem	mbfmptcl 25553*	Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
Theorem	mbfdm2 25554*	The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ dom vol)
Theorem	ismbfcn2 25555*	A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)))
Theorem	ismbfd 25556*	Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 25571. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	ismbf2d 25557*	Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfeqalem1 25558*	Lemma for mbfeqalem2 25559. (Contributed by Mario Carneiro, 2-Sep-2014.) (Revised by AV, 19-Aug-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((◡(𝑥 ∈ 𝐵 ↦ 𝐶) “ 𝑦) ∖ (◡(𝑥 ∈ 𝐵 ↦ 𝐷) “ 𝑦)) ∈ dom vol)
Theorem	mbfeqalem2 25559*	Lemma for mbfeqa 25560. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by AV, 19-Aug-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn))
Theorem	mbfeqa 25560*	If two functions are equal almost everywhere, then one is measurable iff the other is. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn))
Theorem	mbfres 25561	The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ dom vol) → (𝐹 ↾ 𝐴) ∈ MblFn)
Theorem	mbfres2 25562	Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐵 and 𝐶 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn)    &   ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfss 25563*	Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
Theorem	mbfmulc2lem 25564	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)
Theorem	mbfmulc2re 25565	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)
Theorem	mbfmax 25566*	The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐻 ∈ MblFn)
Theorem	mbfneg 25567*	The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
Theorem	mbfpos 25568*	The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
Theorem	mbfposr 25569*	Converse to mbfpos 25568. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
Theorem	mbfposb 25570*	A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))
Theorem	ismbf3d 25571*	Simplified form of ismbfd 25556. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfimaopnlem 25572*	Lemma for mbfimaopn 25573. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝐵 = ((,) “ (ℚ × ℚ))    &   ⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	mbfimaopn 25573	The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 25575, which explains why 𝐴 ∈ dom vol is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	mbfimaopn2 25574	The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐵)    ⇒   ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝐶 ∈ 𝐾) → (◡𝐹 “ 𝐶) ∈ dom vol)
Theorem	cncombf 25575	The composition of a continuous function with a measurable function is measurable. (More generally, 𝐺 can be a Borel-measurable function, but notably the condition that 𝐺 be only measurable is too weak, the usual counterexample taking 𝐺 to be the Cantor function and 𝐹 the indicator function of the 𝐺-image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺 ∈ (𝐵–cn→ℂ)) → (𝐺 ∘ 𝐹) ∈ MblFn)
Theorem	cnmbf 25576	A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐹 ∈ (𝐴–cn→ℂ)) → 𝐹 ∈ MblFn)
Theorem	mbfaddlem 25577	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn)
Theorem	mbfadd 25578	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn)
Theorem	mbfsub 25579	The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ MblFn)
Theorem	mbfmulc2 25580*	A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
Theorem	mbfsup 25581*	The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, 𝐵(𝑛, 𝑥) is a function of both 𝑛 and 𝑥, since it is an 𝑛-indexed sequence of functions on 𝑥. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbfinf 25582*	The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbflimsup 25583*	The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))    &   ⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbflimlem 25584*	The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
Theorem	mbflim 25585*	The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
Syntax	c0p 25586	Extend class notation to include the zero polynomial.
class 0𝑝
Definition	df-0p 25587	Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ 0𝑝 = (ℂ × {0})
Theorem	0pval 25588	The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0)
Theorem	0plef 25589	Two ways to say that the function 𝐹 on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹))
Theorem	0pledm 25590	Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))
Theorem	isi1f 25591	The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because ∫1 is the first preparation function for our final definition ∫ (see df-itg 25540); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
Theorem	i1fmbf 25592	Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn)
Theorem	i1ff 25593	A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
Theorem	i1frn 25594	A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
Theorem	i1fima 25595	Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	i1fima2 25596	Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴) → (vol‘(◡𝐹 “ 𝐴)) ∈ ℝ)
Theorem	i1fima2sn 25597	Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(◡𝐹 “ {𝐴})) ∈ ℝ)
Theorem	i1fd 25598*	A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → ran 𝐹 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
Theorem	i1f0rn 25599	Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
Theorem	itg1val 25600*	The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))))