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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremivthlem1 23201* Lemma for ivth 23204. 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 23202* Lemma for ivth 23204. 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 23203* Lemma for ivth 23204, 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 23204* 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 23205* 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 23206* 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 23207* 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 23208* 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 23209* 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 23210* Combine ivthicc 23208 with evthicc 23209 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 23211* 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 23212 Extend class notation with the outer Lebesgue measure.
class vol*

Syntaxcvol 23213 Extend class notation with the Lebesgue measure.
class vol

Definitiondf-ovol 23214* 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 23215* 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 23216 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 23217* 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 23218* 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 23219 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 23220 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 23221 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))

Theoremovolsf 23222 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 23223* 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*, < ))

Theoremelovolm 23224* 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 23225* 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 23226* 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 23227 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)

Theoremovollb 23228 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 23229* 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 23230 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))

Theoremovolf 23231 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[,]+∞)

Theoremovollecl 23232 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 23233* Lemma for ovolss 23234. (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 23234 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 23235 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 23236 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)

Theoremovollb2lem 23237* Lemma for ovollb2 23238. (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 23238 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 23228). (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 23239 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 23240 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘ℚ) = 0

Theoremovolctb2 23241 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)

Theoremovol0 23242 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
(vol*‘∅) = 0

Theoremovolfi 23243 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)

Theoremovolsn 23244 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)

Theoremovolunlem1a 23245* Lemma for ovolun 23248. (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 23246* Lemma for ovolun 23248. (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 23247 Lemma for ovolun 23248. (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 23248 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 23254, 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 23249 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 23250* 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 23251* Lemma for ovoliun 23254. (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))))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿)       (𝜑 → (𝑈𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))

Theoremovoliunlem2 23252* Lemma for ovoliun 23254. (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 23253* Lemma for ovoliun 23254. (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 23254* 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 23234, 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 23255* 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 23254.) (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 23256* 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 23257* 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 23258* Lemma for ovolshft 23260. (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 23259* Lemma for ovolshft 23260. (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 23260* 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 23261* 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 23262* Lemma for ovolsca 23264. (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 23263* Lemma for ovolshft 23260. (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 23264* 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 23265* 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 23266* Lemma for ovolicc2 23271. (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 23267* Lemma for ovolicc2 23271. (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 23268* Lemma for ovolicc2 23271. (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 23269* Lemma for ovolicc2 23271. (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 23270* Lemma for ovolicc2 23271. (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 23271* 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 23272 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵𝐴))

Theoremovolicopnf 23273 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)

Theoremovolre 23274 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
(vol*‘ℝ) = +∞

Theoremismbl 23275* 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 23276* From ovolun 23248, 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 23277 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol = (vol* ↾ dom vol)

Theoremvolf 23278 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
vol:dom vol⟶(0[,]+∞)

Theoremmblvol 23279 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 23280 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
(𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)

Theoremmblsplit 23281 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 23282 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 23283 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐴 ∈ dom vol → (ℝ ∖ 𝐴) ∈ dom vol)

Theoremnulmbl 23284 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol)

Theoremnulmbl2 23285* 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 23286 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremshftmbl 23287* 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 23288 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
∅ ∈ dom vol

Theoremrembl 23289 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
ℝ ∈ dom vol

Theoremunidmvol 23290 The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)
dom vol = ℝ

Theoreminmbl 23291 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremdifmbl 23292 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)

Theoremfiniunmbl 23293* 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 23294 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 23295 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 23296* 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 23297* 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 23298* 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 23299* Lemma for voliun 23303. (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))

Theoremvoliunlem2 23300* Lemma for voliun 23303. (Contributed by Mario Carneiro, 20-Mar-2014.)
(𝜑𝐹:ℕ⟶dom vol)    &   (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))    &   𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))       (𝜑 ran 𝐹 ∈ dom vol)

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