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Type | Label | Description |
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Statement | ||
Theorem | ovolfs2 24101 | 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 24102 | 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 24103 | 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 24104* | 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 24105* | 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 24106* | 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 24107* | 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 24108 | 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 24109* | 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 24084.) 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 24110* | 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 24084.) (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 24111* | Lemma for uniioombl 24119. (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 24112* | Lemma for uniioombl 24119. (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 24113* | Lemma for uniioombl 24119. (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 24114* | Lemma for uniioombl 24119. (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 24115* | Lemma for uniioombl 24119. (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 24116* | Lemma for uniioombl 24119. (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 24117* | Lemma for uniioombl 24119. (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 24118* | Lemma for uniioombl 24119. (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 24119* | A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24083.) 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 24120* | An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 24083.) (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 24121* | 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 24122* | 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 24123* | 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 24124* | 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 24125* | Lemma for dyaddisj 24126. (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 24126* | 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 24127* | Lemma for dyadmax 24128. (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 24128* | 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 24129* | Lemma for dyadmbl 24130. (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 24130* | 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 24131* | Lemma for opnmbl 24132. (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 24132 | All open sets are measurable. This proof, via dyadmbl 24130 and uniioombl 24119, 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 24133 | All open sets are measurable. This alternative proof of opnmbl 24132 is significantly shorter, at the expense of invoking countable choice ax-cc 9846. (This was also the original proof before the current opnmbl 24132 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 24134 | 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 24135* | 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 24136* | 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 24137* | 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 24138* | Lemma for vitali 24143. (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 24139* | Lemma for vitali 24143. (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 24140* | Lemma for vitali 24143. (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 24141* | Lemma for vitali 24143. (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 24142* | Lemma for vitali 24143. (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 24143 | 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 24144 | Extend class notation with the class of measurable functions. |
class MblFn | ||
Syntax | citg1 24145 | Extend class notation with the Lebesgue integral for simple functions. |
class ∫1 | ||
Syntax | citg2 24146 | Extend class notation with the Lebesgue integral for nonnegative functions. |
class ∫2 | ||
Syntax | cibl 24147 | Extend class notation with the class of integrable functions. |
class 𝐿1 | ||
Syntax | citg 24148 | Extend class notation with the general Lebesgue integral. |
class ∫𝐴𝐵 d𝑥 | ||
Definition | df-mbf 24149* | 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 24056) 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 24150* | 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 24151* | 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 24152* | 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 24153* | 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 24151 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 24151 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 24154* | The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24158 and ismbfcn 24159 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 24155 | 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 24156 | The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | ||
Theorem | mbfconstlem 24157 | Lemma for mbfconst 24163 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) | ||
Theorem | ismbf 24158* | 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 24056. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | ||
Theorem | ismbfcn 24159 | 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 24160 | 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 24161 | 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 24162 | 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 24163 | A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) | ||
Theorem | mbf0 24164 | The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.) |
⊢ ∅ ∈ MblFn | ||
Theorem | mbfid 24165 | The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) | ||
Theorem | mbfmptcl 24166* | 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 24167* | 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 24168* | 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 24169* | 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 24184. (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 24170* | 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 24171* | Lemma for mbfeqalem2 24172. (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 24172* | Lemma for mbfeqa 24173. (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 24173* | 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 24174 | 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 24175 | 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 24176* | 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 24177 | 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 24178 | 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 24179* | 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 24180* | 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 24181* | 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 24182* | Converse to mbfpos 24181. (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 24183* | 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 24184* | Simplified form of ismbfd 24169. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfimaopnlem 24185* | Lemma for mbfimaopn 24186. (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 24186 | The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 24188, 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 24187 | 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 24188 | 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 24189 | 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 24190 | 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 24191 | 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 24192 | 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 24193* | 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 24194* | 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 24195* | 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 24196* | 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 24197* | 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 24198* | 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 24199 | Extend class notation to include the zero polynomial. |
class 0𝑝 | ||
Definition | df-0p 24200 | Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ 0𝑝 = (ℂ × {0}) |
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