The wikipedia article on absolute continuity gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:

$\mu<<\lambda$ if and only if for every $\epsilon>0$ and for every bounded real interval $I$ there is a $\delta>0$ such that for every (finite or infinite) sequence of pairwise disjoint sub-intervals [$x_i,y_i$] of $I$ with

$\sum_{i} |y_i - x_i| < \delta$

it follows that

$\sum_{i} |\mu((-\infty, y_i])-\mu((-\infty, x_i])| < \epsilon$.

My questions are: Does this connection generalise? What would be a topological reformulation of $\mu<<\nu$? If the notion does not generalise for arbitrary measures, does it generalise for the Lebesgue measure on $R^n$? How would a sketch of the proof look like?