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Definition df-itg2 21123
Description: Define the Lebesgue integral for nonnegative functions. A nonnegative function's integral is the supremum of the integrals of all simple functions that are less than the input function. Note that this may be +oo for functions that take the value +oo on a set of positive measure or functions that are bounded below by a positive number on a set of infinite measure. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
df-itg2  |-  S.2  =  ( f  e.  ( ( 0 [,] +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
Distinct variable group:    f, g, x

Detailed syntax breakdown of Definition df-itg2
StepHypRef Expression
1 citg2 21118 . 2  class  S.2
2 vf . . 3  setvar  f
3 cc0 9303 . . . . 5  class  0
4 cpnf 9436 . . . . 5  class +oo
5 cicc 11324 . . . . 5  class  [,]
63, 4, 5co 6112 . . . 4  class  ( 0 [,] +oo )
7 cr 9302 . . . 4  class  RR
8 cmap 7235 . . . 4  class  ^m
96, 7, 8co 6112 . . 3  class  ( ( 0 [,] +oo )  ^m  RR )
10 vg . . . . . . . . 9  setvar  g
1110cv 1368 . . . . . . . 8  class  g
122cv 1368 . . . . . . . 8  class  f
13 cle 9440 . . . . . . . . 9  class  <_
1413cofr 6340 . . . . . . . 8  class  oR  <_
1511, 12, 14wbr 4313 . . . . . . 7  wff  g  oR  <_  f
16 vx . . . . . . . . 9  setvar  x
1716cv 1368 . . . . . . . 8  class  x
18 citg1 21117 . . . . . . . . 9  class  S.1
1911, 18cfv 5439 . . . . . . . 8  class  ( S.1 `  g )
2017, 19wceq 1369 . . . . . . 7  wff  x  =  ( S.1 `  g
)
2115, 20wa 369 . . . . . 6  wff  ( g  oR  <_  f  /\  x  =  ( S.1 `  g ) )
2218cdm 4861 . . . . . 6  class  dom  S.1
2321, 10, 22wrex 2737 . . . . 5  wff  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g ) )
2423, 16cab 2429 . . . 4  class  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) }
25 cxr 9438 . . . 4  class  RR*
26 clt 9439 . . . 4  class  <
2724, 25, 26csup 7711 . . 3  class  sup ( { x  |  E. g  e.  dom  S.1 (
g  oR  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  )
282, 9, 27cmpt 4371 . 2  class  ( f  e.  ( ( 0 [,] +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 (
g  oR  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  ) )
291, 28wceq 1369 1  wff  S.2  =  ( f  e.  ( ( 0 [,] +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
Colors of variables: wff setvar class
This definition is referenced by:  itg2val  21228
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