Some questions in measure theory. I got two questions.
Suppose we have the statement

If a property Q holds a.e, then P holds a.e.

Wouldd the contrapositive be,

If ~P a.e, then ~Q a.e.

Or would we remove a.e and say it holds everywhere?
Also can someone check if this argument is valid (I've seen the proof elsewhere, but this is the one I wrote).

If $f,g$ are continuous functions on $[a,b]$ with $f = g$ a.e, then in fact $f = g$.

I basically said let $x \in \{ x : f \neq g \}$ and WLOG, let $f - g > 0$.
Setting $h = f - g$, and for small $\delta$, we get $h(x + \delta) - h(x) > 0$. Passing the limit gives $0 > 0$ (by continuity), a contradiction.

You should insert quantifiers: "$P$ holds a.e." can be written as
$$(\exists N \in \mathcal{A}) \: \mu(N)=0 \wedge \left [ (\forall x \in X \setminus N) \: P(x) \right ].$$
So the negation of that can be written as
$$(\forall N \in \mathcal{A}) \: \mu(N) \neq 0 \vee \left [ (\exists x \in X \setminus N) \: \neg P(x) \right ].$$
A slightly nicer form:
$$(\forall N \in \mathcal{A}) \: \mu(N) = 0 \Rightarrow \left [ (\exists x \in X \setminus N) \: \neg P(x) \right ].$$
The intuitive meaning is "the property fails on a set of positive measure".
