Definition df-mbfm 34585

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 ℝ to ℂ, see df-mbf 25747. (Contributed by Thierry Arnoux, 23-Jan-2017.)

Assertion

Ref	Expression
df-mbfm	⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

Distinct variable group:   𝑓,𝑠,𝑡,𝑥

Detailed syntax breakdown of Definition df-mbfm

Step	Hyp	Ref	Expression
1		cmbfm 34584	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 34443	. . . . 5 class sigAlgebra
5	4	crn 5663	. . . 4 class ran sigAlgebra
6	5	cuni 4874	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1566	. . . . . . . 8 class 𝑓
9	8	ccnv 5661	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1566	. . . . . . 7 class 𝑥
12	9, 11	cima 5665	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1566	. . . . . 6 class 𝑠
14	12, 13	wcel 2149	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1566	. . . . 5 class 𝑡
16	14, 10, 15	wral 3085	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4874	. . . . 5 class ∪ 𝑡
18	13	cuni 4874	. . . . 5 class ∪ 𝑠
19		cmap 8824	. . . . 5 class ↑m
20	17, 18, 19	co 7411	. . . 4 class (∪ 𝑡 ↑m ∪ 𝑠)
21	16, 7, 20	crab 3422	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpo 7413	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1567	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  34586  elunirnmbfm  34587