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Definition df-vol 23440
Description: 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.)
Assertion
Ref Expression
df-vol vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-vol
StepHypRef Expression
1 cvol 23438 . 2 class vol
2 covol 23437 . . 3 class vol*
3 vy . . . . . . . 8 setvar 𝑦
43cv 1636 . . . . . . 7 class 𝑦
54, 2cfv 6095 . . . . . 6 class (vol*‘𝑦)
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1636 . . . . . . . . 9 class 𝑥
84, 7cin 3762 . . . . . . . 8 class (𝑦𝑥)
98, 2cfv 6095 . . . . . . 7 class (vol*‘(𝑦𝑥))
104, 7cdif 3760 . . . . . . . 8 class (𝑦𝑥)
1110, 2cfv 6095 . . . . . . 7 class (vol*‘(𝑦𝑥))
12 caddc 10218 . . . . . . 7 class +
139, 11, 12co 6868 . . . . . 6 class ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))
145, 13wceq 1637 . . . . 5 wff (vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))
152ccnv 5304 . . . . . 6 class vol*
16 cr 10214 . . . . . 6 class
1715, 16cima 5308 . . . . 5 class (vol* “ ℝ)
1814, 3, 17wral 3092 . . . 4 wff 𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))
1918, 6cab 2788 . . 3 class {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))}
202, 19cres 5307 . 2 class (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
211, 20wceq 1637 1 wff vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
Colors of variables: wff setvar class
This definition is referenced by:  ismbl  23501  volres  23503
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