Definition df-mbfm 33248

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 25136. (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 33247	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 33106	. . . . 5 class sigAlgebra
5	4	crn 5678	. . . 4 class ran sigAlgebra
6	5	cuni 4909	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1541	. . . . . . . 8 class 𝑓
9	8	ccnv 5676	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1541	. . . . . . 7 class 𝑥
12	9, 11	cima 5680	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1541	. . . . . 6 class 𝑠
14	12, 13	wcel 2107	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1541	. . . . 5 class 𝑡
16	14, 10, 15	wral 3062	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4909	. . . . 5 class ∪ 𝑡
18	13	cuni 4909	. . . . 5 class ∪ 𝑠
19		cmap 8820	. . . . 5 class ↑m
20	17, 18, 19	co 7409	. . . 4 class (∪ 𝑡 ↑m ∪ 𝑠)
21	16, 7, 20	crab 3433	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpo 7411	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1542	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  33249  elunirnmbfm  33250