Let $X$ = {0}. Use the trivial measure on $X$ such that $\mu(A)$ = 0 for all measurable sets in $X$. Let the property $P$ be such that $P$ does not hold at 0. But then there exists a set $N = X$ with $\mu(N) = 0$ and all $x \in X - N = \emptyset$ have the property $P$ (this is true since there is no such $x$). Thus, it appears that $P$ holds almost everywhere on $X$. But, simultaneously, there is no $x \in X$ such that $P$ holds, so we might say $P$ holds nowhere.
Is then correct that a property can hold almost everywhere and also nowhere as in the above example, or did I misunderstand something?