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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ovolfsf 24001 | Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) ⇒ ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) | ||
Theorem | ovolsf 24002 | Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) & ⊢ 𝑆 = seq1( + , 𝐺) ⇒ ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) | ||
Theorem | ovolval 24003* | The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.) |
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) | ||
Theorem | elovolmlem 24004 | Lemma for elovolm 24005 and related theorems. (Contributed by BJ, 23-Jul-2022.) |
⊢ (𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ))) | ||
Theorem | elovolm 24005* | Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) | ||
Theorem | elovolmr 24006* | Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇒ ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀) | ||
Theorem | ovolmge0 24007* | The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵) | ||
Theorem | ovolcl 24008 | The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | ||
Theorem | ovollb 24009 | The outer volume is a lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.) |
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇒ ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < )) | ||
Theorem | ovolgelb 24010* | The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.) |
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⇒ ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵))) | ||
Theorem | ovolge0 24011 | The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴)) | ||
Theorem | ovolf 24012 | The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | ||
Theorem | ovollecl 24013 | If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (vol*‘𝐴) ≤ 𝐵) → (vol*‘𝐴) ∈ ℝ) | ||
Theorem | ovolsslem 24014* | Lemma for ovolss 24015. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} & ⊢ 𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) | ||
Theorem | ovolss 24015 | The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) | ||
Theorem | ovolsscl 24016 | If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ) | ||
Theorem | ovolssnul 24017 | A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0) | ||
Theorem | ovollb2lem 24018* | Lemma for ovollb2 24019. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉) & ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)) | ||
Theorem | ovollb2 24019 | It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 24009). (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇒ ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < )) | ||
Theorem | ovolctb 24020 | The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0) | ||
Theorem | ovolq 24021 | The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘ℚ) = 0 | ||
Theorem | ovolctb2 24022 | The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) | ||
Theorem | ovol0 24023 | The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘∅) = 0 | ||
Theorem | ovolfi 24024 | A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0) | ||
Theorem | ovolsn 24025 | A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | ||
Theorem | ovolunlem1a 24026* | Lemma for ovolun 24029. (Contributed by Mario Carneiro, 7-May-2015.) |
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) & ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) & ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) & ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) & ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) & ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2)))) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) | ||
Theorem | ovolunlem1 24027* | Lemma for ovolun 24029. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) & ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) & ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) & ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) & ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) & ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2)))) ⇒ ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) | ||
Theorem | ovolunlem2 24028 | Lemma for ovolun 24029. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) & ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) | ||
Theorem | ovolun 24029 | The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 24035, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) | ||
Theorem | ovolunnul 24030 | Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴)) | ||
Theorem | ovolfiniun 24031* | The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)) | ||
Theorem | ovoliunlem1 24032* | Lemma for ovoliun 24035. (Contributed by Mario Carneiro, 12-Jun-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ 𝑇 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ) & ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) & ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻)) & ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘)))) & ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ)) & ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛))) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽‘𝑤)) ≤ 𝐿) ⇒ ⊢ (𝜑 → (𝑈‘𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)) | ||
Theorem | ovoliunlem2 24033* | Lemma for ovoliun 24035. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ 𝑇 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ) & ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) & ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻)) & ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘)))) & ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ)) & ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛))) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) ⇒ ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)) | ||
Theorem | ovoliunlem3 24034* | Lemma for ovoliun 24035. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ 𝑇 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ) & ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)) | ||
Theorem | ovoliun 24035* | The Lebesgue outer measure function is countably sub-additive. (Many books allow +∞ as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 24015, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ 𝑇 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )) | ||
Theorem | ovoliun2 24036* | The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 24035.) (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ 𝑇 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝑇 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴)) | ||
Theorem | ovoliunnul 24037* | A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.) |
⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0) | ||
Theorem | shft2rab 24038* | If 𝐵 is a shift of 𝐴 by 𝐶, then 𝐴 is a shift of 𝐵 by -𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) ⇒ ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵}) | ||
Theorem | ovolshftlem1 24039* | Lemma for ovolshft 24041. (Contributed by Mario Carneiro, 22-Mar-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) & ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)〉) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) ⇒ ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀) | ||
Theorem | ovolshftlem2 24040* | Lemma for ovolshft 24041. (Contributed by Mario Carneiro, 22-Mar-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) & ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀) | ||
Theorem | ovolshft 24041* | The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) ⇒ ⊢ (𝜑 → (vol*‘𝐴) = (vol*‘𝐵)) | ||
Theorem | sca2rab 24042* | If 𝐵 is a scale of 𝐴 by 𝐶, then 𝐴 is a scale of 𝐵 by 1 / 𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) ⇒ ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵}) | ||
Theorem | ovolscalem1 24043* | Lemma for ovolsca 24045. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) & ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)〉) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))) ⇒ ⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)) | ||
Theorem | ovolscalem2 24044* | Lemma for ovolshft 24041. (Contributed by Mario Carneiro, 22-Mar-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) & ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶)) | ||
Theorem | ovolsca 24045* | The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) & ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶)) | ||
Theorem | ovolicc1 24046* | The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 〈𝐴, 𝐵〉, 〈0, 0〉)) ⇒ ⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴)) | ||
Theorem | ovolicc2lem1 24047* | Lemma for ovolicc2 24052. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝐺:𝑈⟶ℕ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋)))))) | ||
Theorem | ovolicc2lem2 24048* | Lemma for ovolicc2 24052. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝐺:𝑈⟶ℕ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) & ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} & ⊢ (𝜑 → 𝐻:𝑇⟶𝑇) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡)) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶})) & ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)} ⇒ ⊢ ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁 ∈ 𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) ≤ 𝐵) | ||
Theorem | ovolicc2lem3 24049* | Lemma for ovolicc2 24052. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝐺:𝑈⟶ℕ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) & ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} & ⊢ (𝜑 → 𝐻:𝑇⟶𝑇) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡)) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶})) & ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)} ⇒ ⊢ ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃)))))) | ||
Theorem | ovolicc2lem4 24050* | Lemma for ovolicc2 24052. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝐺:𝑈⟶ℕ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) & ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} & ⊢ (𝜑 → 𝐻:𝑇⟶𝑇) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡)) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶})) & ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)} & ⊢ 𝑀 = inf(𝑊, ℝ, < ) ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )) | ||
Theorem | ovolicc2lem5 24051* | Lemma for ovolicc2 24052. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) & ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) & ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝐺:𝑈⟶ℕ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) & ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )) | ||
Theorem | ovolicc2 24052* | The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵))) | ||
Theorem | ovolicc 24053 | The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)) | ||
Theorem | ovolicopnf 24054 | The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞) | ||
Theorem | ovolre 24055 | The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (vol*‘ℝ) = +∞ | ||
Theorem | ismbl 24056* | 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 24057* | From ovolun 24029, 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 24058 | A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ vol = (vol* ↾ dom vol) | ||
Theorem | volf 24059 | The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ vol:dom vol⟶(0[,]+∞) | ||
Theorem | mblvol 24060 | 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 24061 | A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | ||
Theorem | mblsplit 24062 | 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 24063 | 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 24064 | The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ (𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol) | ||
Theorem | nulmbl 24065 | A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) | ||
Theorem | nulmbl2 24066* | 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 24067 | 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 24068* | 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 24069 | The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ∅ ∈ dom vol | ||
Theorem | rembl 24070 | The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ℝ ∈ dom vol | ||
Theorem | unidmvol 24071 | The union of the Lebesgue measurable sets is ℝ. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ ∪ dom vol = ℝ | ||
Theorem | inmbl 24072 | 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 24073 | 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 24074* | 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 24075 | 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 24076 | 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 24077* | 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 24078* | 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 24079* | 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 24080* | Lemma for voliun 24084. (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 24081* | Lemma for voliun 24084. (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 24082* | Lemma for voliun 24084. (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 24083 | 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 24084 | 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 24085* | Lemma for volsup 24086. (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 24086* | 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 24087* | 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 24088* | Lemma for ioombl1 24092. (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 24089* | Lemma for ioombl1 24092. (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 24090* | Lemma for ioombl1 24092. (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 24091* | Lemma for ioombl1 24092. (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 24092 | 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 24093 | A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol) | ||
Theorem | icombl 24094 | 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 24095 | An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ (𝐴(,)𝐵) ∈ dom vol | ||
Theorem | iccmbl 24096 | A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol) | ||
Theorem | iccvolcl 24097 | A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,]𝐵)) ∈ ℝ) | ||
Theorem | ovolioo 24098 | The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) | ||
Theorem | volioo 24099 | The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) | ||
Theorem | ioovolcl 24100 | An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ) |
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