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Definition df-mbfm 28580
Description: Define the measurable function builder, which generates the set of measurable functions from a measurable space to another one. Here, the measurable spaces are given using their sigma algebra  s and  t, and the spaces themselves are recovered by  U. s and  U. t.

Note the similarities between the definition of measurable functions in measure theory, and of continuous functions in topology.

This is the definition for the generic measure theory. For the specific case of functions from  RR to  CC, see df-mbf 22210 (Contributed by Thierry Arnoux, 23-Jan-2017.)

Assertion
Ref Expression
df-mbfm  |- MblFnM  =  ( s  e.  U. ran sigAlgebra , 
t  e.  U. ran sigAlgebra  |->  { f  e.  ( U. t  ^m  U. s )  |  A. x  e.  t  ( `' f
" x )  e.  s } )
Distinct variable group:    f, s, t, x

Detailed syntax breakdown of Definition df-mbfm
StepHypRef Expression
1 cmbfm 28579 . 2  class MblFnM
2 vs . . 3  setvar  s
3 vt . . 3  setvar  t
4 csiga 28436 . . . . 5  class sigAlgebra
54crn 4941 . . . 4  class  ran sigAlgebra
65cuni 4188 . . 3  class  U. ran sigAlgebra
7 vf . . . . . . . . 9  setvar  f
87cv 1402 . . . . . . . 8  class  f
98ccnv 4939 . . . . . . 7  class  `' f
10 vx . . . . . . . 8  setvar  x
1110cv 1402 . . . . . . 7  class  x
129, 11cima 4943 . . . . . 6  class  ( `' f " x )
132cv 1402 . . . . . 6  class  s
1412, 13wcel 1840 . . . . 5  wff  ( `' f " x )  e.  s
153cv 1402 . . . . 5  class  t
1614, 10, 15wral 2751 . . . 4  wff  A. x  e.  t  ( `' f " x )  e.  s
1715cuni 4188 . . . . 5  class  U. t
1813cuni 4188 . . . . 5  class  U. s
19 cmap 7375 . . . . 5  class  ^m
2017, 18, 19co 6232 . . . 4  class  ( U. t  ^m  U. s )
2116, 7, 20crab 2755 . . 3  class  { f  e.  ( U. t  ^m  U. s )  | 
A. x  e.  t  ( `' f "
x )  e.  s }
222, 3, 6, 6, 21cmpt2 6234 . 2  class  ( s  e.  U. ran sigAlgebra ,  t  e.  U. ran sigAlgebra  |->  { f  e.  ( U. t  ^m  U. s )  | 
A. x  e.  t  ( `' f "
x )  e.  s } )
231, 22wceq 1403 1  wff MblFnM  =  ( s  e.  U. ran sigAlgebra , 
t  e.  U. ran sigAlgebra  |->  { f  e.  ( U. t  ^m  U. s )  |  A. x  e.  t  ( `' f
" x )  e.  s } )
Colors of variables: wff setvar class
This definition is referenced by:  ismbfm  28581  elunirnmbfm  28582
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