Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that (a) assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a>b$ and (b) assigns an infinitesimal--ideally, the same one--to each singleton?

It's (a) that's a problem.  The Bernstein-Wattenberg construction yields a finitely-additive measure that gives (a) up to infinitesimals.  But it would be nice to have (a) exactly.