![]() |
Metamath
Proof Explorer Theorem List (p. 257 of 486) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30851) |
![]() (30852-32374) |
![]() (32375-48553) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | uniioombllem1 25601* | Lemma for uniioombl 25609. (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 25602* | Lemma for uniioombl 25609. (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 25603* | Lemma for uniioombl 25609. (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 25604* | Lemma for uniioombl 25609. (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 25605* | Lemma for uniioombl 25609. (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 25606* | Lemma for uniioombl 25609. (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 25607* | Lemma for uniioombl 25609. (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 25608* | Lemma for uniioombl 25609. (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 25609* | A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25573.) 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 25610* | An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25573.) (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 25611* | 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 25612* | 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 25613* | 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 25614* | 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 25615* | Lemma for dyaddisj 25616. (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 25616* | 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 25617* | Lemma for dyadmax 25618. (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 25618* | 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 25619* | Lemma for dyadmbl 25620. (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 25620* | 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 25621* | Lemma for opnmbl 25622. (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 25622 | All open sets are measurable. This proof, via dyadmbl 25620 and uniioombl 25609, 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 25623 | All open sets are measurable. This alternative proof of opnmbl 25622 is significantly shorter, at the expense of invoking countable choice ax-cc 10478. (This was also the original proof before the current opnmbl 25622 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 25624 | 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 25625* | 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 25626* | 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 25627* | 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 25628* | Lemma for vitali 25633. (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 25629* | Lemma for vitali 25633. (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 25630* | Lemma for vitali 25633. (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 25631* | Lemma for vitali 25633. (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 25632* | Lemma for vitali 25633. (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 25633 | 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) | ||
Syntax | cmbf 25634 | Extend class notation with the class of measurable functions. |
class MblFn | ||
Syntax | citg1 25635 | Extend class notation with the Lebesgue integral for simple functions. |
class ∫1 | ||
Syntax | citg2 25636 | Extend class notation with the Lebesgue integral for nonnegative functions. |
class ∫2 | ||
Syntax | cibl 25637 | Extend class notation with the class of integrable functions. |
class 𝐿1 | ||
Syntax | citg 25638 | Extend class notation with the general Lebesgue integral. |
class ∫𝐴𝐵 d𝑥 | ||
Definition | df-mbf 25639* | 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 25546) 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 25640* | 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 25641* | 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 25642* | 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 25643* | 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 25641 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 25641 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 25644* | The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25648 and ismbfcn 25649 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 25645 | 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 25646 | The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | ||
Theorem | mbfconstlem 25647 | Lemma for mbfconst 25653 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) | ||
Theorem | ismbf 25648* | 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 25546. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | ||
Theorem | ismbfcn 25649 | 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 25650 | 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 25651 | 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 25652 | 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 25653 | A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) | ||
Theorem | mbf0 25654 | The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.) |
⊢ ∅ ∈ MblFn | ||
Theorem | mbfid 25655 | The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) | ||
Theorem | mbfmptcl 25656* | 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 25657* | 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 25658* | 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 25659* | 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 25674. (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 25660* | 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 25661* | Lemma for mbfeqalem2 25662. (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 25662* | Lemma for mbfeqa 25663. (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 25663* | 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 25664 | 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 25665 | 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 25666* | 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 25667 | 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 25668 | 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 25669* | 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 25670* | 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 25671* | 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 25672* | Converse to mbfpos 25671. (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 25673* | 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 25674* | Simplified form of ismbfd 25659. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfimaopnlem 25675* | Lemma for mbfimaopn 25676. (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 25676 | The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 25678, 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 25677 | 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 25678 | 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 25679 | 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 25680 | 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 25681 | 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 25682 | 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 25683* | 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 25684* | 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 25685* | 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 25686* | 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 25687* | 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 25688* | 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 25689 | Extend class notation to include the zero polynomial. |
class 0𝑝 | ||
Definition | df-0p 25690 | Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ 0𝑝 = (ℂ × {0}) | ||
Theorem | 0pval 25691 | The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | ||
Theorem | 0plef 25692 | 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 25693 | 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 25694 | 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 25643); 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 25695 | Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | ||
Theorem | i1ff 25696 | A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | ||
Theorem | i1frn 25697 | A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | ||
Theorem | i1fima 25698 | Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) | ||
Theorem | i1fima2 25699 | 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 25700 | Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(◡𝐹 “ {𝐴})) ∈ ℝ) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |