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Definition df-itg1 18976
Description: Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
df-itg1  |-  S.1  =  ( f  e.  {
g  e. MblFn  |  (
g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR ) }  |->  sum_
x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f " {
x } ) ) ) )
Distinct variable group:    f, g, x

Detailed syntax breakdown of Definition df-itg1
StepHypRef Expression
1 citg1 18970 . 2  class  S.1
2 vf . . 3  set  f
3 cr 8736 . . . . . 6  class  RR
4 vg . . . . . . 7  set  g
54cv 1622 . . . . . 6  class  g
63, 3, 5wf 5251 . . . . 5  wff  g : RR --> RR
75crn 4690 . . . . . 6  class  ran  g
8 cfn 6863 . . . . . 6  class  Fin
97, 8wcel 1684 . . . . 5  wff  ran  g  e.  Fin
105ccnv 4688 . . . . . . . 8  class  `' g
11 cc0 8737 . . . . . . . . . 10  class  0
1211csn 3640 . . . . . . . . 9  class  { 0 }
133, 12cdif 3149 . . . . . . . 8  class  ( RR 
\  { 0 } )
1410, 13cima 4692 . . . . . . 7  class  ( `' g " ( RR 
\  { 0 } ) )
15 cvol 18823 . . . . . . 7  class  vol
1614, 15cfv 5255 . . . . . 6  class  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )
1716, 3wcel 1684 . . . . 5  wff  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR
186, 9, 17w3a 934 . . . 4  wff  ( g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR )
19 cmbf 18969 . . . 4  class MblFn
2018, 4, 19crab 2547 . . 3  class  { g  e. MblFn  |  ( g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  e.  RR ) }
212cv 1622 . . . . . 6  class  f
2221crn 4690 . . . . 5  class  ran  f
2322, 12cdif 3149 . . . 4  class  ( ran  f  \  { 0 } )
24 vx . . . . . 6  set  x
2524cv 1622 . . . . 5  class  x
2621ccnv 4688 . . . . . . 7  class  `' f
2725csn 3640 . . . . . . 7  class  { x }
2826, 27cima 4692 . . . . . 6  class  ( `' f " { x } )
2928, 15cfv 5255 . . . . 5  class  ( vol `  ( `' f " { x } ) )
30 cmul 8742 . . . . 5  class  x.
3125, 29, 30co 5858 . . . 4  class  ( x  x.  ( vol `  ( `' f " {
x } ) ) )
3223, 31, 24csu 12158 . . 3  class  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )
332, 20, 32cmpt 4077 . 2  class  ( f  e.  { g  e. MblFn  |  ( g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  e.  RR ) }  |->  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) ) )
341, 33wceq 1623 1  wff  S.1  =  ( f  e.  {
g  e. MblFn  |  (
g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR ) }  |->  sum_
x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f " {
x } ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  isi1f  19029  itg1val  19038
  Copyright terms: Public domain W3C validator