measure theory - What is the "continuity" in "absolute continuity", in general?

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?

• Next ct.category theory - Iterated adjoint functors
• Previous ag.algebraic geometry - Why can't subvarieties separate?