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Definition df-mbfm 32263
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 24828. (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 32262 . 2 class MblFnM
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 csiga 32121 . . . . 5 class sigAlgebra
54crn 5601 . . . 4 class ran sigAlgebra
65cuni 4844 . . 3 class ran sigAlgebra
7 vf . . . . . . . . 9 setvar 𝑓
87cv 1538 . . . . . . . 8 class 𝑓
98ccnv 5599 . . . . . . 7 class 𝑓
10 vx . . . . . . . 8 setvar 𝑥
1110cv 1538 . . . . . . 7 class 𝑥
129, 11cima 5603 . . . . . 6 class (𝑓𝑥)
132cv 1538 . . . . . 6 class 𝑠
1412, 13wcel 2104 . . . . 5 wff (𝑓𝑥) ∈ 𝑠
153cv 1538 . . . . 5 class 𝑡
1614, 10, 15wral 3062 . . . 4 wff 𝑥𝑡 (𝑓𝑥) ∈ 𝑠
1715cuni 4844 . . . . 5 class 𝑡
1813cuni 4844 . . . . 5 class 𝑠
19 cmap 8646 . . . . 5 class m
2017, 18, 19co 7307 . . . 4 class ( 𝑡m 𝑠)
2116, 7, 20crab 3284 . . 3 class {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠}
222, 3, 6, 6, 21cmpo 7309 . 2 class (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
231, 22wceq 1539 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  32264  elunirnmbfm  32265
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