Definition df-mbfm 30313

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 23388. (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 30312	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 30170	. . . . 5 class sigAlgebra
5	4	crn 5115	. . . 4 class ran sigAlgebra
6	5	cuni 4436	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1482	. . . . . . . 8 class 𝑓
9	8	ccnv 5113	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1482	. . . . . . 7 class 𝑥
12	9, 11	cima 5117	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1482	. . . . . 6 class 𝑠
14	12, 13	wcel 1990	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1482	. . . . 5 class 𝑡
16	14, 10, 15	wral 2912	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4436	. . . . 5 class ∪ 𝑡
18	13	cuni 4436	. . . . 5 class ∪ 𝑠
19		cmap 7857	. . . . 5 class ↑𝑚
20	17, 18, 19	co 6650	. . . 4 class (∪ 𝑡 ↑𝑚 ∪ 𝑠)
21	16, 7, 20	crab 2916	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpt2 6652	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1483	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  30314  elunirnmbfm  30315