Definition df-mbfm 30441

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 23433. (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 30440	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 30298	. . . . 5 class sigAlgebra
5	4	crn 5144	. . . 4 class ran sigAlgebra
6	5	cuni 4468	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1522	. . . . . . . 8 class 𝑓
9	8	ccnv 5142	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1522	. . . . . . 7 class 𝑥
12	9, 11	cima 5146	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1522	. . . . . 6 class 𝑠
14	12, 13	wcel 2030	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1522	. . . . 5 class 𝑡
16	14, 10, 15	wral 2941	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4468	. . . . 5 class ∪ 𝑡
18	13	cuni 4468	. . . . 5 class ∪ 𝑠
19		cmap 7899	. . . . 5 class ↑𝑚
20	17, 18, 19	co 6690	. . . 4 class (∪ 𝑡 ↑𝑚 ∪ 𝑠)
21	16, 7, 20	crab 2945	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpt2 6692	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1523	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  30442  elunirnmbfm  30443