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.

First, I don't thing you can say that WLOG, let $f - g > 0$.  I don't see any reason why they both couldn't oscillate and switch relative positions very often.  However even if $f - g > 0$ you still have a problem with $h(x + \delta) - h(x) > 0$.  In your case $h(x + \delta) > 0$ and $h(x) > 0$, but I don't see why the difference would have to be positive.
