Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that 

 1. assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and 

 2. assigns an infinitesimal--ideally, the same one--to each singleton?

It's (1) that's a problem.  The Bernstein-Wattenberg construction yields a finitely-additive measure that gives (1) up to infinitesimals.  But it would be nice to have (1) exactly.