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Definition df-mbfm 26830
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 21235 (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 26829 . 2  class MblFnM
2 vs . . 3  setvar  s
3 vt . . 3  setvar  t
4 csiga 26715 . . . . 5  class sigAlgebra
54crn 4952 . . . 4  class  ran sigAlgebra
65cuni 4202 . . 3  class  U. ran sigAlgebra
7 vf . . . . . . . . 9  setvar  f
87cv 1369 . . . . . . . 8  class  f
98ccnv 4950 . . . . . . 7  class  `' f
10 vx . . . . . . . 8  setvar  x
1110cv 1369 . . . . . . 7  class  x
129, 11cima 4954 . . . . . 6  class  ( `' f " x )
132cv 1369 . . . . . 6  class  s
1412, 13wcel 1758 . . . . 5  wff  ( `' f " x )  e.  s
153cv 1369 . . . . 5  class  t
1614, 10, 15wral 2799 . . . 4  wff  A. x  e.  t  ( `' f " x )  e.  s
1715cuni 4202 . . . . 5  class  U. t
1813cuni 4202 . . . . 5  class  U. s
19 cmap 7327 . . . . 5  class  ^m
2017, 18, 19co 6203 . . . 4  class  ( U. t  ^m  U. s )
2116, 7, 20crab 2803 . . 3  class  { f  e.  ( U. t  ^m  U. s )  | 
A. x  e.  t  ( `' f "
x )  e.  s }
222, 3, 6, 6, 21cmpt2 6205 . 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 1370 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  26831  elunirnmbfm  26832
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