Definition df-mbfm 33859

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 25541. (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 33858	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 33717	. . . . 5 class sigAlgebra
5	4	crn 5673	. . . 4 class ran sigAlgebra
6	5	cuni 4903	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1533	. . . . . . . 8 class 𝑓
9	8	ccnv 5671	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1533	. . . . . . 7 class 𝑥
12	9, 11	cima 5675	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1533	. . . . . 6 class 𝑠
14	12, 13	wcel 2099	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1533	. . . . 5 class 𝑡
16	14, 10, 15	wral 3056	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4903	. . . . 5 class ∪ 𝑡
18	13	cuni 4903	. . . . 5 class ∪ 𝑠
19		cmap 8838	. . . . 5 class ↑m
20	17, 18, 19	co 7414	. . . 4 class (∪ 𝑡 ↑m ∪ 𝑠)
21	16, 7, 20	crab 3427	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpo 7416	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1534	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  33860  elunirnmbfm  33861