Definition df-mbfm

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 U. s and U. 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 RR to CC , see df-mbf . (Contributed by Thierry Arnoux, 23-Jan-2017)

	Ref	Expression
Assertion	df-mbfm	⊢ MblFnM = ( 𝑠 ∈ ∪ ran sigAlgebra , 𝑡 ∈ ∪ ran sigAlgebra ↦ { 𝑓 ∈ ( ∪ 𝑡 ↑m ∪ 𝑠 ) ∣ ∀ 𝑥 ∈ 𝑡 ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠 } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cmbfm	⊢ MblFnM
1		vs	⊢ 𝑠
2		csiga	⊢ sigAlgebra
3	2	crn	⊢ ran sigAlgebra
4	3	cuni	⊢ ∪ ran sigAlgebra
5		vt	⊢ 𝑡
6		vf	⊢ 𝑓
7	5	cv	⊢ 𝑡
8	7	cuni	⊢ ∪ 𝑡
9		cmap	⊢ ↑m
10	1	cv	⊢ 𝑠
11	10	cuni	⊢ ∪ 𝑠
12	8 11 9	co	⊢ ( ∪ 𝑡 ↑m ∪ 𝑠 )
13		vx	⊢ 𝑥
14	6	cv	⊢ 𝑓
15	14	ccnv	⊢ ◡ 𝑓
16	13	cv	⊢ 𝑥
17	15 16	cima	⊢ ( ◡ 𝑓 “ 𝑥 )
18	17 10	wcel	⊢ ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠
19	18 13 7	wral	⊢ ∀ 𝑥 ∈ 𝑡 ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠
20	19 6 12	crab	⊢ { 𝑓 ∈ ( ∪ 𝑡 ↑m ∪ 𝑠 ) ∣ ∀ 𝑥 ∈ 𝑡 ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠 }
21	1 5 4 4 20	cmpo	⊢ ( 𝑠 ∈ ∪ ran sigAlgebra , 𝑡 ∈ ∪ ran sigAlgebra ↦ { 𝑓 ∈ ( ∪ 𝑡 ↑m ∪ 𝑠 ) ∣ ∀ 𝑥 ∈ 𝑡 ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠 } )
22	0 21	wceq	⊢ MblFnM = ( 𝑠 ∈ ∪ ran sigAlgebra , 𝑡 ∈ ∪ ran sigAlgebra ↦ { 𝑓 ∈ ( ∪ 𝑡 ↑m ∪ 𝑠 ) ∣ ∀ 𝑥 ∈ 𝑡 ( ◡ 𝑓 “ 𝑥 ) ∈ 𝑠 } )