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Definition df-mbfm 30086
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 23289. (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 30085 . 2 class MblFnM
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 csiga 29943 . . . . 5 class sigAlgebra
54crn 5080 . . . 4 class ran sigAlgebra
65cuni 4407 . . 3 class ran sigAlgebra
7 vf . . . . . . . . 9 setvar 𝑓
87cv 1479 . . . . . . . 8 class 𝑓
98ccnv 5078 . . . . . . 7 class 𝑓
10 vx . . . . . . . 8 setvar 𝑥
1110cv 1479 . . . . . . 7 class 𝑥
129, 11cima 5082 . . . . . 6 class (𝑓𝑥)
132cv 1479 . . . . . 6 class 𝑠
1412, 13wcel 1992 . . . . 5 wff (𝑓𝑥) ∈ 𝑠
153cv 1479 . . . . 5 class 𝑡
1614, 10, 15wral 2912 . . . 4 wff 𝑥𝑡 (𝑓𝑥) ∈ 𝑠
1715cuni 4407 . . . . 5 class 𝑡
1813cuni 4407 . . . . 5 class 𝑠
19 cmap 7803 . . . . 5 class 𝑚
2017, 18, 19co 6605 . . . 4 class ( 𝑡𝑚 𝑠)
2116, 7, 20crab 2916 . . 3 class {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠}
222, 3, 6, 6, 21cmpt2 6607 . 2 class (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
231, 22wceq 1480 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  30087  elunirnmbfm  30088
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