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 𝑚 ∖ 𝑎 ) ) = 0 } |
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 𝑚 ∖ 𝑎 ) |
8 | 7 3 | cfv | ⊢ ( 𝑚 ‘ ( ∪ dom 𝑚 ∖ 𝑎 ) ) |
9 | cc0 | ⊢ 0 | |
10 | 8 9 | wceq | ⊢ ( 𝑚 ‘ ( ∪ dom 𝑚 ∖ 𝑎 ) ) = 0 |
11 | 10 1 2 | copab | ⊢ { 〈 𝑎 , 𝑚 〉 ∣ ( 𝑚 ‘ ( ∪ dom 𝑚 ∖ 𝑎 ) ) = 0 } |
12 | 0 11 | wceq | ⊢ a.e. = { 〈 𝑎 , 𝑚 〉 ∣ ( 𝑚 ‘ ( ∪ dom 𝑚 ∖ 𝑎 ) ) = 0 } |