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Definition df-ovol 22957
Description: 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.)
Assertion
Ref Expression
df-ovol vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
Distinct variable group:   𝑥,𝑦,𝑓

Detailed syntax breakdown of Definition df-ovol
StepHypRef Expression
1 covol 22955 . 2 class vol*
2 vx . . 3 setvar 𝑥
3 cr 9791 . . . 4 class
43cpw 4107 . . 3 class 𝒫 ℝ
52cv 1473 . . . . . . . 8 class 𝑥
6 cioo 12002 . . . . . . . . . . 11 class (,)
7 vf . . . . . . . . . . . 12 setvar 𝑓
87cv 1473 . . . . . . . . . . 11 class 𝑓
96, 8ccom 5032 . . . . . . . . . 10 class ((,) ∘ 𝑓)
109crn 5029 . . . . . . . . 9 class ran ((,) ∘ 𝑓)
1110cuni 4366 . . . . . . . 8 class ran ((,) ∘ 𝑓)
125, 11wss 3539 . . . . . . 7 wff 𝑥 ran ((,) ∘ 𝑓)
13 vy . . . . . . . . 9 setvar 𝑦
1413cv 1473 . . . . . . . 8 class 𝑦
15 caddc 9795 . . . . . . . . . . 11 class +
16 cabs 13768 . . . . . . . . . . . . 13 class abs
17 cmin 10117 . . . . . . . . . . . . 13 class
1816, 17ccom 5032 . . . . . . . . . . . 12 class (abs ∘ − )
1918, 8ccom 5032 . . . . . . . . . . 11 class ((abs ∘ − ) ∘ 𝑓)
20 c1 9793 . . . . . . . . . . 11 class 1
2115, 19, 20cseq 12618 . . . . . . . . . 10 class seq1( + , ((abs ∘ − ) ∘ 𝑓))
2221crn 5029 . . . . . . . . 9 class ran seq1( + , ((abs ∘ − ) ∘ 𝑓))
23 cxr 9929 . . . . . . . . 9 class *
24 clt 9930 . . . . . . . . 9 class <
2522, 23, 24csup 8206 . . . . . . . 8 class sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )
2614, 25wceq 1474 . . . . . . 7 wff 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )
2712, 26wa 382 . . . . . 6 wff (𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
28 cle 9931 . . . . . . . 8 class
293, 3cxp 5026 . . . . . . . 8 class (ℝ × ℝ)
3028, 29cin 3538 . . . . . . 7 class ( ≤ ∩ (ℝ × ℝ))
31 cn 10867 . . . . . . 7 class
32 cmap 7721 . . . . . . 7 class 𝑚
3330, 31, 32co 6527 . . . . . 6 class (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)
3427, 7, 33wrex 2896 . . . . 5 wff 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
3534, 13, 23crab 2899 . . . 4 class {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
3635, 23, 24cinf 8207 . . 3 class inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < )
372, 4, 36cmpt 4637 . 2 class (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
381, 37wceq 1474 1 wff vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
Colors of variables: wff setvar class
This definition is referenced by:  ovolval  22966  ovolf  22974
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