Metamath Proof Explorer


Definition df-sitg

Description: Define the integral of simple functions from a measurable space dom m to a generic space w equipped with the right scalar product. w will later be required to be a Banach space.

These simple functions are required to take finitely many different values: this is expressed by ran g e. Fin in the definition.

Moreover, for each x , the pre-image (`' g " { x } ) is requested to be measurable, of finite measure.

In this definition, ( sigaGen `( TopOpenw ) ) is the Borel sigma-algebra on w , and the functions g range over the measurable functions over that Borel algebra.

Definition 2.4.1 of Bogachev p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017)

Ref Expression
Assertion df-sitg
|- sitg = ( w e. _V , m e. U. ran measures |-> ( f e. { g e. ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsum ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 csitg
 |-  sitg
1 vw
 |-  w
2 cvv
 |-  _V
3 vm
 |-  m
4 cmeas
 |-  measures
5 4 crn
 |-  ran measures
6 5 cuni
 |-  U. ran measures
7 vf
 |-  f
8 vg
 |-  g
9 3 cv
 |-  m
10 9 cdm
 |-  dom m
11 cmbfm
 |-  MblFnM
12 csigagen
 |-  sigaGen
13 ctopn
 |-  TopOpen
14 1 cv
 |-  w
15 14 13 cfv
 |-  ( TopOpen ` w )
16 15 12 cfv
 |-  ( sigaGen ` ( TopOpen ` w ) )
17 10 16 11 co
 |-  ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) )
18 8 cv
 |-  g
19 18 crn
 |-  ran g
20 cfn
 |-  Fin
21 19 20 wcel
 |-  ran g e. Fin
22 vx
 |-  x
23 c0g
 |-  0g
24 14 23 cfv
 |-  ( 0g ` w )
25 24 csn
 |-  { ( 0g ` w ) }
26 19 25 cdif
 |-  ( ran g \ { ( 0g ` w ) } )
27 18 ccnv
 |-  `' g
28 22 cv
 |-  x
29 28 csn
 |-  { x }
30 27 29 cima
 |-  ( `' g " { x } )
31 30 9 cfv
 |-  ( m ` ( `' g " { x } ) )
32 cc0
 |-  0
33 cico
 |-  [,)
34 cpnf
 |-  +oo
35 32 34 33 co
 |-  ( 0 [,) +oo )
36 31 35 wcel
 |-  ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo )
37 36 22 26 wral
 |-  A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo )
38 21 37 wa
 |-  ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) )
39 38 8 17 crab
 |-  { g e. ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) }
40 cgsu
 |-  gsum
41 7 cv
 |-  f
42 41 crn
 |-  ran f
43 42 25 cdif
 |-  ( ran f \ { ( 0g ` w ) } )
44 crrh
 |-  RRHom
45 csca
 |-  Scalar
46 14 45 cfv
 |-  ( Scalar ` w )
47 46 44 cfv
 |-  ( RRHom ` ( Scalar ` w ) )
48 41 ccnv
 |-  `' f
49 48 29 cima
 |-  ( `' f " { x } )
50 49 9 cfv
 |-  ( m ` ( `' f " { x } ) )
51 50 47 cfv
 |-  ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) )
52 cvsca
 |-  .s
53 14 52 cfv
 |-  ( .s ` w )
54 51 28 53 co
 |-  ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x )
55 22 43 54 cmpt
 |-  ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) )
56 14 55 40 co
 |-  ( w gsum ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) )
57 7 39 56 cmpt
 |-  ( f e. { g e. ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsum ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) )
58 1 3 2 6 57 cmpo
 |-  ( w e. _V , m e. U. ran measures |-> ( f e. { g e. ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsum ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) )
59 0 58 wceq
 |-  sitg = ( w e. _V , m e. U. ran measures |-> ( f e. { g e. ( dom m MblFnM ( sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsum ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( ( RRHom ` ( Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) )