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Type | Label | Description |
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Statement | ||
Theorem | ivthlem1 25201* | Lemma for ivth 25204. The set 𝑆 of all 𝑥 values with (𝐹‘𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)) | ||
Theorem | ivthlem2 25202* | Lemma for ivth 25204. Show that the supremum of 𝑆 cannot be less than 𝑈. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than 𝑈, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} & ⊢ 𝐶 = sup(𝑆, ℝ, < ) ⇒ ⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈) | ||
Theorem | ivthlem3 25203* | Lemma for ivth 25204, the intermediate value theorem. Show that (𝐹‘𝐶) cannot be greater than 𝑈, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} & ⊢ 𝐶 = sup(𝑆, ℝ, < ) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹‘𝐶) = 𝑈)) | ||
Theorem | ivth 25204* | The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivth2 25205* | The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthle 25206* | The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐵))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthle2 25207* | The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐴))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈) | ||
Theorem | ivthicc 25208* | The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑁 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐹‘𝑀)[,](𝐹‘𝑁)) ⊆ ran 𝐹) | ||
Theorem | evthicc 25209* | Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤))) | ||
Theorem | evthicc2 25210* | Combine ivthicc 25208 with evthicc 25209 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦)) | ||
Theorem | cniccbdd 25211* | A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑥) | ||
Syntax | covol 25212 | Extend class notation with the outer Lebesgue measure. |
class vol* | ||
Syntax | cvol 25213 | Extend class notation with the Lebesgue measure. |
class vol | ||
Definition | df-ovol 25214* | Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs ⟨𝑎, 𝑏⟩ with 𝑎 ≤ 𝑏, and the outer volume of the set 𝑥 is the infimum over all such functions such that the union of the open intervals (𝑎, 𝑏) covers 𝑥 of the sum of 𝑏 − 𝑎. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.) |
⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < )) | ||
Definition | df-vol 25215* | Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as 𝐴 ∈ dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) | ||
Theorem | ovolfcl 25216 | Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁)))) | ||
Theorem | ovolfioo 25217* | Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑛))))) | ||
Theorem | ovolficc 25218* | Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑛))))) | ||
Theorem | ovolficcss 25219 | Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) | ||
Theorem | ovolfsval 25220 | The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) ⇒ ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) | ||
Theorem | ovolfsf 25221 | 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 25222 | 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 25223* | 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 25224 | Lemma for elovolm 25225 and related theorems. (Contributed by BJ, 23-Jul-2022.) |
⊢ (𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ))) | ||
Theorem | elovolm 25225* | 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 25226* | 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 25227* | 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 25228 | The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | ||
Theorem | ovollb 25229 | 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 25230* | 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 25231 | The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴)) | ||
Theorem | ovolf 25232 | The domain and codomain 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 25233 | 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 25234* | Lemma for ovolss 25235. (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 25235 | 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 25236 | 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 25237 | 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 25238* | Lemma for ovollb2 25239. (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 25239 | 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 25229). (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 25240 | 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 25241 | The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘ℚ) = 0 | ||
Theorem | ovolctb2 25242 | The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) | ||
Theorem | ovol0 25243 | The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘∅) = 0 | ||
Theorem | ovolfi 25244 | A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0) | ||
Theorem | ovolsn 25245 | A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | ||
Theorem | ovolunlem1a 25246* | Lemma for ovolun 25249. (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 25247* | Lemma for ovolun 25249. (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 25248 | Lemma for ovolun 25249. (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 25249 | The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 25255, 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 25250 | 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 25251* | 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 25252* | Lemma for ovoliun 25255. (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 25253* | Lemma for ovoliun 25255. (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 25254* | Lemma for ovoliun 25255. (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 25255* | 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 25235, 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 25256* | 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 25255.) (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 25257* | 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 25258* | 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 25259* | Lemma for ovolshft 25261. (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 25260* | Lemma for ovolshft 25261. (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 25261* | 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 25262* | 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 25263* | Lemma for ovolsca 25265. (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 25264* | Lemma for ovolshft 25261. (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 25265* | 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 25266* | 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 25267* | Lemma for ovolicc2 25272. (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 25268* | Lemma for ovolicc2 25272. (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 25269* | Lemma for ovolicc2 25272. (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 25270* | Lemma for ovolicc2 25272. (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 25271* | Lemma for ovolicc2 25272. (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 25272* | 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 25273 | 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 25274 | The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞) | ||
Theorem | ovolre 25275 | The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (vol*‘ℝ) = +∞ | ||
Theorem | ismbl 25276* | 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 25277* | From ovolun 25249, 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 25278 | A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ vol = (vol* ↾ dom vol) | ||
Theorem | volf 25279 | The domain and codomain of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ vol:dom vol⟶(0[,]+∞) | ||
Theorem | mblvol 25280 | 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 25281 | A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | ||
Theorem | mblsplit 25282 | 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 25283 | 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 25284 | The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ (𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol) | ||
Theorem | nulmbl 25285 | A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) | ||
Theorem | nulmbl2 25286* | 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 25287 | 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 25288* | 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 25289 | The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ∅ ∈ dom vol | ||
Theorem | rembl 25290 | The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ ℝ ∈ dom vol | ||
Theorem | unidmvol 25291 | The union of the Lebesgue measurable sets is ℝ. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ ∪ dom vol = ℝ | ||
Theorem | inmbl 25292 | 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 25293 | 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 25294* | 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 25295 | 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 25296 | 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 25297* | 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 25298* | 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 25299* | 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 25300* | Lemma for voliun 25304. (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)) |
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