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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremivthlem1 25201* Lemma for ivth 25204. The set 𝑆 of all 𝑥 values with (𝐹𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}       (𝜑 → (𝐴𝑆 ∧ ∀𝑧𝑆 𝑧𝐵))
 
Theoremivthlem2 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → ¬ (𝐹𝐶) < 𝑈)
 
Theoremivthlem3 25203* Lemma for ivth 25204, the intermediate value theorem. Show that (𝐹𝐶) 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹𝐶) = 𝑈))
 
Theoremivth 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivth2 25205* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) < 𝑈𝑈 < (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle 25206* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) ≤ 𝑈𝑈 ≤ (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthle2 25207* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) ≤ 𝑈𝑈 ≤ (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)
 
Theoremivthicc 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ (𝐴[,]𝐵))    &   (𝜑𝑁 ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)       (𝜑 → ((𝐹𝑀)[,](𝐹𝑁)) ⊆ ran 𝐹)
 
Theoremevthicc 25209* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹𝑧) ≤ (𝐹𝑤)))
 
Theoremevthicc2 25210* Combine ivthicc 25208 with evthicc 25209 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦))
 
Theoremcniccbdd 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‘(𝐹𝑦)) ≤ 𝑥)
 
13.2  Integrals
 
13.2.1  Lebesgue measure
 
Syntaxcovol 25212 Extend class notation with the outer Lebesgue measure.
class vol*
 
Syntaxcvol 25213 Extend class notation with the Lebesgue measure.
class vol
 
Definitiondf-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 𝑎𝑏, 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 𝑏𝑎. (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*, < ))
 
Definitiondf-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*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
 
Theoremovolfcl 25216 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
 
Theoremovolfioo 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) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
 
Theoremovolficc 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) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
 
Theoremovolficcss 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)
 
Theoremovolfsval 25220 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
 
Theoremovolfsf 25221 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
 
Theoremovolsf 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[,)+∞))
 
Theoremovolval 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*, < ))
 
Theoremelovolmlem 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)))
 
Theoremelovolm 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*, < ))}       (𝐵𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
 
Theoremelovolmr 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)
 
Theoremovolmge0 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*, < ))}       (𝐵𝑀 → 0 ≤ 𝐵)
 
Theoremovolcl 25228 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
 
Theoremovollb 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*, < ))
 
Theoremovolgelb 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)))
 
Theoremovolge0 25231 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
 
Theoremovolf 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[,]+∞)
 
Theoremovollecl 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)
 
Theoremovolsslem 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*, < ))}       ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolss 25235 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 
Theoremovolsscl 25236 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ)
 
Theoremovolssnul 25237 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)
 
Theoremovollb2lem 25238* Lemma for ovollb2 25239. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ([,] ∘ 𝐹))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)       (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
 
Theoremovollb2 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*, < ))
 
Theoremovolctb 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)
 
Theoremovolq 25241 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘ℚ) = 0
 
Theoremovolctb2 25242 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)
 
Theoremovol0 25243 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘∅) = 0
 
Theoremovolfi 25244 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)
 
Theoremovolsn 25245 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
 
Theoremovolunlem1a 25246* Lemma for ovolun 25249. (Contributed by Mario Carneiro, 7-May-2015.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   (𝜑𝐵 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))))       ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolunlem1 25247* Lemma for ovolun 25249. (Contributed by Mario Carneiro, 12-Jun-2014.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   (𝜑𝐵 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*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolunlem2 25248 Lemma for ovolun 25249. (Contributed by Mario Carneiro, 12-Jun-2014.)
(𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
 
Theoremovolun 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*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
 
Theoremovolunnul 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*‘(𝐴𝐵)) = (vol*‘𝐴))
 
Theoremovolfiniun 25251* The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵))
 
Theoremovoliunlem1 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))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))    &   (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))    &   ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿)       (𝜑 → (𝑈𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliunlem2 25253* Lemma for ovoliun 25255. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))    &   𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))    &   (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))    &   ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliunlem3 25254* Lemma for ovoliun 25255. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑇 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 
Theoremovoliun 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))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
 
Theoremovoliun2 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))    &   ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   (𝜑𝑇 ∈ dom ⇝ )       (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
 
Theoremovoliunnul 25257* A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
 
Theoremshft2rab 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.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})       (𝜑𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵})
 
Theoremovolshftlem1 25259* Lemma for ovolshft 25261. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 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 ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))       (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 
Theoremovolshftlem2 25260* Lemma for ovolshft 25261. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 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)
 
Theoremovolshft 25261* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})       (𝜑 → (vol*‘𝐴) = (vol*‘𝐵))
 
Theoremsca2rab 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.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})       (𝜑𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵})
 
Theoremovolscalem1 25263* Lemma for ovolsca 25265. (Contributed by Mario Carneiro, 6-Apr-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ((,) ∘ 𝐹))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))       (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
 
Theoremovolscalem2 25264* Lemma for ovolshft 25261. (Contributed by Mario Carneiro, 22-Mar-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶))
 
Theoremovolsca 25265* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   (𝜑 → (vol*‘𝐴) ∈ ℝ)       (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶))
 
Theoremovolicc1 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))       (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
 
Theoremovolicc2lem1 25267* Lemma for ovolicc2 25272. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)       ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
 
Theoremovolicc2lem2 25268* Lemma for ovolicc2 25272. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}       ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵)
 
Theoremovolicc2lem3 25269* Lemma for ovolicc2 25272. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}       ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃))))))
 
Theoremovolicc2lem4 25270* Lemma for ovolicc2 25272. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   (𝜑𝐻:𝑇𝑇)    &   ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝑇)    &   𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}    &   𝑀 = inf(𝑊, ℝ, < )       (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolicc2lem5 25271* Lemma for ovolicc2 25272. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝐺:𝑈⟶ℕ)    &   ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)    &   𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}       (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 
Theoremovolicc2 25272* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}       (𝜑 → (𝐵𝐴) ≤ (vol*‘(𝐴[,]𝐵)))
 
Theoremovolicc 25273 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵𝐴))
 
Theoremovolicopnf 25274 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)
 
Theoremovolre 25275 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
(vol*‘ℝ) = +∞
 
Theoremismbl 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 𝑥𝐴 and 𝑥𝐴 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*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
 
Theoremismbl2 25277* From ovolun 25249, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥𝐴 and 𝑥𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
 
Theoremvolres 25278 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol = (vol* ↾ dom vol)
 
Theoremvolf 25279 The domain and codomain of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol:dom vol⟶(0[,]+∞)
 
Theoremmblvol 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))
 
Theoremmblss 25281 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
 
Theoremmblsplit 25282 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) = ((vol*‘(𝐵𝐴)) + (vol*‘(𝐵𝐴))))
 
Theoremvolss 25283 The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴𝐵) → (vol‘𝐴) ≤ (vol‘𝐵))
 
Theoremcmmbl 25284 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)
 
Theoremnulmbl 25285 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)
 
Theoremnulmbl2 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(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
 
Theoremunmbl 25287 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremshftmbl 25288* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ) → {𝑥 ∈ ℝ ∣ (𝑥𝐵) ∈ 𝐴} ∈ dom vol)
 
Theorem0mbl 25289 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
∅ ∈ dom vol
 
Theoremrembl 25290 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
ℝ ∈ dom vol
 
Theoremunidmvol 25291 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)
dom vol = ℝ
 
Theoreminmbl 25292 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremdifmbl 25293 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
 
Theoremfiniunmbl 25294* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ dom vol) → 𝑘𝐴 𝐵 ∈ dom vol)
 
Theoremvolun 25295 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
(((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))
 
Theoremvolinun 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‘(𝐴𝐵)) + (vol‘(𝐴𝐵))))
 
Theoremvolfiniun 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 ∧ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))
 
Theoremiundisj 25298* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremiundisj2 25299* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
(𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremvoliunlem1 25300* Lemma for voliun 25304. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝜑𝐹:ℕ⟶dom vol)    &   (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))    &   𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))    &   (𝜑𝐸 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)       ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
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