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Definition df-mbfm 34251
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-algebras 𝑠 and 𝑡, and the spaces themselves are recovered by 𝑠 and 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 to , see df-mbf 25654. (Contributed by Thierry Arnoux, 23-Jan-2017.)

Assertion
Ref Expression
df-mbfm MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
Distinct variable group:   𝑓,𝑠,𝑡,𝑥

Detailed syntax breakdown of Definition df-mbfm
StepHypRef Expression
1 cmbfm 34250 . 2 class MblFnM
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 csiga 34109 . . . . 5 class sigAlgebra
54crn 5686 . . . 4 class ran sigAlgebra
65cuni 4907 . . 3 class ran sigAlgebra
7 vf . . . . . . . . 9 setvar 𝑓
87cv 1539 . . . . . . . 8 class 𝑓
98ccnv 5684 . . . . . . 7 class 𝑓
10 vx . . . . . . . 8 setvar 𝑥
1110cv 1539 . . . . . . 7 class 𝑥
129, 11cima 5688 . . . . . 6 class (𝑓𝑥)
132cv 1539 . . . . . 6 class 𝑠
1412, 13wcel 2108 . . . . 5 wff (𝑓𝑥) ∈ 𝑠
153cv 1539 . . . . 5 class 𝑡
1614, 10, 15wral 3061 . . . 4 wff 𝑥𝑡 (𝑓𝑥) ∈ 𝑠
1715cuni 4907 . . . . 5 class 𝑡
1813cuni 4907 . . . . 5 class 𝑠
19 cmap 8866 . . . . 5 class m
2017, 18, 19co 7431 . . . 4 class ( 𝑡m 𝑠)
2116, 7, 20crab 3436 . . 3 class {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠}
222, 3, 6, 6, 21cmpo 7433 . 2 class (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
231, 22wceq 1540 1 wff MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
Colors of variables: wff setvar class
This definition is referenced by:  ismbfm  34252  elunirnmbfm  34253
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