Some subtle questions in measure theory This is the definition of 'almost everywhere' in Folland Real Analysis. 

If $(X,\mathcal{M},\mu)$ is a measure space, a set $E \in \mathcal{M}$ such that $\mu(E) = 0$ is called a null set.  By subadditivity, any countable union of null sets is a null set, a fact which we shall use frequently.  If a statement about points $x\in X$ is true except for $x$ in some null set, we say that it is true almost everywhere (abbreviated a.e.), or for almost every $x$.  (If more precision is needed, we shall speak of a $\mu$-null set, or $mu$-almost everywhere.)

But, there is something ambiguous in the def.
What I understand is that a statement on X is almost everywhere true if the subset of X in which the given statement is not true is a 'null-set' itself. It seems to be true in the context...but the folland book is written somewhat ambiguously..
Am I correct?
 A: You are not completely correct.
Ordinarily, one says (and I interpret Folland as meaning this) that a property $E(x)$ is valid ($\mu$-)almost everywhere, if there is a ($\mu$-)null set $N \in \mathcal{M}$ such that $E(x)$ is true for all $x \in X \setminus N$.
If the set $M := \{x \in X\mid E(x) \text{ is true}\}$ is measurable, then this is the same as saying that $M^c$ is a null-set.
But it can happen that $M$ is not measurable, although $E(x)$ holds almost everywhere.
For a (trivial) example, consider $X = \Bbb{N}$, $\mathcal{M} = \{\emptyset, X\}$ and $\mu \equiv 0$. Then every property holds almost everywhere (we can take $N = X$), but if $E(x)$ means (e.g.) that $x$ is even, then $M$ is not measurable.
This is somewhat related to the concept of a complete measure space (this is treated in Folland), which is a measure space with the property that if $N \in \mathcal{M}$ has measure $0$ and if $M \subset N$, then $M \in \mathcal{M}$.
The measure space given above is not complete.
In complete measure spaces, both formulations are easily seen to be equivalent.
