Suppose we have Lebesgue measurable set E. Let F be a subset of E with measure zero with respect to the Lebesgue measure on E. My question is, can we construct a "reasonable" integration theory on the set F that can be extended to the set E ? In other words, what are the alternative theory of integrations on a set of measure zero (the measure zero sets are with respect to Lebesgue measure ) ?
On many sets of measure zero, you can construct the "Hausdorff measure." The Cantor set is a good example of such a set. http://en.wikipedia.org/wiki/Hausdorff_measure
For countable sets like the rationals, I think measures absolutely continuous with respect to counting measure are all you are going to get.