Definition df-mbfm 30653

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 23607. (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 30652	. 2 class MblFnM
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		csiga 30510	. . . . 5 class sigAlgebra
5	4	crn 5250	. . . 4 class ran sigAlgebra
6	5	cuni 4574	. . 3 class ∪ ran sigAlgebra
7		vf	. . . . . . . . 9 setvar 𝑓
8	7	cv 1630	. . . . . . . 8 class 𝑓
9	8	ccnv 5248	. . . . . . 7 class ◡𝑓
10		vx	. . . . . . . 8 setvar 𝑥
11	10	cv 1630	. . . . . . 7 class 𝑥
12	9, 11	cima 5252	. . . . . 6 class (◡𝑓 “ 𝑥)
13	2	cv 1630	. . . . . 6 class 𝑠
14	12, 13	wcel 2145	. . . . 5 wff (◡𝑓 “ 𝑥) ∈ 𝑠
15	3	cv 1630	. . . . 5 class 𝑡
16	14, 10, 15	wral 3061	. . . 4 wff ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠
17	15	cuni 4574	. . . . 5 class ∪ 𝑡
18	13	cuni 4574	. . . . 5 class ∪ 𝑠
19		cmap 8009	. . . . 5 class ↑𝑚
20	17, 18, 19	co 6793	. . . 4 class (∪ 𝑡 ↑𝑚 ∪ 𝑠)
21	16, 7, 20	crab 3065	. . 3 class {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}
22	2, 3, 6, 6, 21	cmpt2 6795	. 2 class (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
23	1, 22	wceq 1631	1 wff MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})

This definition is referenced by:  ismbfm  30654  elunirnmbfm  30655