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Definition df-mbfm 30653
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 23607. (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 30652 . 2 class MblFnM
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 csiga 30510 . . . . 5 class sigAlgebra
54crn 5250 . . . 4 class ran sigAlgebra
65cuni 4574 . . 3 class ran sigAlgebra
7 vf . . . . . . . . 9 setvar 𝑓
87cv 1630 . . . . . . . 8 class 𝑓
98ccnv 5248 . . . . . . 7 class 𝑓
10 vx . . . . . . . 8 setvar 𝑥
1110cv 1630 . . . . . . 7 class 𝑥
129, 11cima 5252 . . . . . 6 class (𝑓𝑥)
132cv 1630 . . . . . 6 class 𝑠
1412, 13wcel 2145 . . . . 5 wff (𝑓𝑥) ∈ 𝑠
153cv 1630 . . . . 5 class 𝑡
1614, 10, 15wral 3061 . . . 4 wff 𝑥𝑡 (𝑓𝑥) ∈ 𝑠
1715cuni 4574 . . . . 5 class 𝑡
1813cuni 4574 . . . . 5 class 𝑠
19 cmap 8009 . . . . 5 class 𝑚
2017, 18, 19co 6793 . . . 4 class ( 𝑡𝑚 𝑠)
2116, 7, 20crab 3065 . . 3 class {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠}
222, 3, 6, 6, 21cmpt2 6795 . 2 class (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
231, 22wceq 1631 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  30654  elunirnmbfm  30655
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