Metamath Proof Explorer


Definition df-ae

Description: Define 'almost everywhere' with regard to a measure M . A property holds almost everywhere if the measure of the set where it does not hold has measure zero. (Contributed by Thierry Arnoux, 20-Oct-2017)

Ref Expression
Assertion df-ae a.e. = { ⟨ 𝑎 , 𝑚 ⟩ ∣ ( 𝑚 ‘ ( dom 𝑚𝑎 ) ) = 0 }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cae a.e.
1 va 𝑎
2 vm 𝑚
3 2 cv 𝑚
4 3 cdm dom 𝑚
5 4 cuni dom 𝑚
6 1 cv 𝑎
7 5 6 cdif ( dom 𝑚𝑎 )
8 7 3 cfv ( 𝑚 ‘ ( dom 𝑚𝑎 ) )
9 cc0 0
10 8 9 wceq ( 𝑚 ‘ ( dom 𝑚𝑎 ) ) = 0
11 10 1 2 copab { ⟨ 𝑎 , 𝑚 ⟩ ∣ ( 𝑚 ‘ ( dom 𝑚𝑎 ) ) = 0 }
12 0 11 wceq a.e. = { ⟨ 𝑎 , 𝑚 ⟩ ∣ ( 𝑚 ‘ ( dom 𝑚𝑎 ) ) = 0 }