The concept of "almost everywhere" wrt to zero-dimensional measure $\mathcal H^0$ From my understanding, a function $u$ is defined $\mathcal H^0$-a.e. on an interval $I$ means that $u$ is only not "defined" on a set of points $A \subset I$ such that $\mathcal H^0(A)=0$
However, I think $\mathcal H^0(A)=0$ means that $A$ is empty, and hence implies that $u$ is defined everywhere in $I$. Because even if $A$ contains only a point, then $\mathcal H^0(A)=1>0$.
My question: can somebody confirm for me that "defined $\mathcal H^0$ a.e. is equivalent to defined everywhere" is correct? or am I making some trivial mistake?

PS: $\mathcal H^0$ means Housdroff measure in $0$ dimensions. Also, we may think it as a counting measure.

Yes, the zero-dimensional Hausdorff measure $\mathcal H^0$ is simply the counting measure, and the only null set for this measure is the empty set. 
Example of usage: if $A\subset \mathbb{R}^2$ is a Borel set of zero length (meaning $\mathcal H^1(A)=0$), then for almost every $x\in\mathbb{R}$ the slice 
$$A_x = \{y\in \mathbb{R}: (x,y)\in A\}$$
has zero $\mathcal{H}^0$ measure. (This is an instance of a more general result.) In practical terms, this means the intersection of $A$ with almost every vertical line is empty.
