Lebesgue integration may offer a way out for some functions (namely, those with discrete outputs, even if not connected), as there you calculate a weighted sum over the measures of the supports of each output. This is one way to argue that the integral from 0 to 1 of the indicator function of the irrationals is 1, for example. Sadly, while you never quite end up with rectangles of width zero, you do something sufficiently equivalent in general (IIRC).

But I'm not so sure there's a paradox after all: we never actually consider the rectangles of width zero (or the analog in Lebesgue), as we are taking the limit as (something) approaches zero. All the paradox really says is that the summation is discontinuous at width=0, which is possibly less alarming?