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Definition df-mbfm 31623
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 24227. (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 31622 . 2 class MblFnM
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 csiga 31481 . . . . 5 class sigAlgebra
54crn 5524 . . . 4 class ran sigAlgebra
65cuni 4803 . . 3 class ran sigAlgebra
7 vf . . . . . . . . 9 setvar 𝑓
87cv 1537 . . . . . . . 8 class 𝑓
98ccnv 5522 . . . . . . 7 class 𝑓
10 vx . . . . . . . 8 setvar 𝑥
1110cv 1537 . . . . . . 7 class 𝑥
129, 11cima 5526 . . . . . 6 class (𝑓𝑥)
132cv 1537 . . . . . 6 class 𝑠
1412, 13wcel 2112 . . . . 5 wff (𝑓𝑥) ∈ 𝑠
153cv 1537 . . . . 5 class 𝑡
1614, 10, 15wral 3109 . . . 4 wff 𝑥𝑡 (𝑓𝑥) ∈ 𝑠
1715cuni 4803 . . . . 5 class 𝑡
1813cuni 4803 . . . . 5 class 𝑠
19 cmap 8393 . . . . 5 class m
2017, 18, 19co 7139 . . . 4 class ( 𝑡m 𝑠)
2116, 7, 20crab 3113 . . 3 class {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠}
222, 3, 6, 6, 21cmpo 7141 . 2 class (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
231, 22wceq 1538 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  31624  elunirnmbfm  31625
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