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Definition df-vol 18825
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  e.  dom  vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
Assertion
Ref Expression
df-vol  |-  vol  =  ( vol *  |`  { x  |  A. y  e.  ( `' vol * " RR ) ( vol * `  y )  =  ( ( vol * `  ( y  i^i  x
) )  +  ( vol * `  (
y  \  x )
) ) } )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-vol
StepHypRef Expression
1 cvol 18823 . 2  class  vol
2 covol 18822 . . 3  class  vol *
3 vy . . . . . . . 8  set  y
43cv 1622 . . . . . . 7  class  y
54, 2cfv 5255 . . . . . 6  class  ( vol
* `  y )
6 vx . . . . . . . . . 10  set  x
76cv 1622 . . . . . . . . 9  class  x
84, 7cin 3151 . . . . . . . 8  class  ( y  i^i  x )
98, 2cfv 5255 . . . . . . 7  class  ( vol
* `  ( y  i^i  x ) )
104, 7cdif 3149 . . . . . . . 8  class  ( y 
\  x )
1110, 2cfv 5255 . . . . . . 7  class  ( vol
* `  ( y  \  x ) )
12 caddc 8740 . . . . . . 7  class  +
139, 11, 12co 5858 . . . . . 6  class  ( ( vol * `  (
y  i^i  x )
)  +  ( vol
* `  ( y  \  x ) ) )
145, 13wceq 1623 . . . . 5  wff  ( vol
* `  y )  =  ( ( vol
* `  ( y  i^i  x ) )  +  ( vol * `  ( y  \  x
) ) )
152ccnv 4688 . . . . . 6  class  `' vol *
16 cr 8736 . . . . . 6  class  RR
1715, 16cima 4692 . . . . 5  class  ( `' vol * " RR )
1814, 3, 17wral 2543 . . . 4  wff  A. y  e.  ( `' vol * " RR ) ( vol
* `  y )  =  ( ( vol
* `  ( y  i^i  x ) )  +  ( vol * `  ( y  \  x
) ) )
1918, 6cab 2269 . . 3  class  { x  |  A. y  e.  ( `' vol * " RR ) ( vol * `  y )  =  ( ( vol * `  ( y  i^i  x
) )  +  ( vol * `  (
y  \  x )
) ) }
202, 19cres 4691 . 2  class  ( vol
*  |`  { x  | 
A. y  e.  ( `' vol * " RR ) ( vol * `  y )  =  ( ( vol * `  ( y  i^i  x
) )  +  ( vol * `  (
y  \  x )
) ) } )
211, 20wceq 1623 1  wff  vol  =  ( vol *  |`  { x  |  A. y  e.  ( `' vol * " RR ) ( vol * `  y )  =  ( ( vol * `  ( y  i^i  x
) )  +  ( vol * `  (
y  \  x )
) ) } )
Colors of variables: wff set class
This definition is referenced by:  ismbl  18885  volres  18887
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