Metamath Proof Explorer

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 ⊢ MblFn μ = s ∈ ⋃ ran ⁡ sigAlgebra , t ∈ ⋃ ran ⁡ sigAlgebra ⟼ f ∈ ⋃ t ⋃ s | ∀ x ∈ t f -1 x ∈ s

Detailed syntax breakdown

Step Hyp Ref Expression
0 cmbfm class MblFn μ
1 vs setvar s
2 csiga class sigAlgebra
3 2 crn class ran ⁡ sigAlgebra
4 3 cuni class ⋃ ran ⁡ sigAlgebra
5 vt setvar t
6 vf setvar f
7 5 cv setvar t
8 7 cuni class ⋃ t
9 cmap class ↑ 𝑚
10 1 cv setvar s
11 10 cuni class ⋃ s
12 8 11 9 co class ⋃ t ⋃ s
13 vx setvar x
14 6 cv setvar f
15 14 ccnv class f -1
16 13 cv setvar x
17 15 16 cima class f -1 x
18 17 10 wcel wff f -1 x ∈ s
19 18 13 7 wral wff ∀ x ∈ t f -1 x ∈ s
20 19 6 12 crab class f ∈ ⋃ t ⋃ s | ∀ x ∈ t f -1 x ∈ s
21 1 5 4 4 20 cmpo class s ∈ ⋃ ran ⁡ sigAlgebra , t ∈ ⋃ ran ⁡ sigAlgebra ⟼ f ∈ ⋃ t ⋃ s | ∀ x ∈ t f -1 x ∈ s
22 0 21 wceq wff MblFn μ = s ∈ ⋃ ran ⁡ sigAlgebra , t ∈ ⋃ ran ⁡ sigAlgebra ⟼ f ∈ ⋃ t ⋃ s | ∀ x ∈ t f -1 x ∈ s