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.



Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
Yes, by compactness. 


Let $R$ denote your favorite hyperreal ordered field 
and let $\delta\in R$ be a positive infinitesimal. 
Let $\mathcal{E}$ denote the set of all 
(standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$.
For every $A\in\mathcal{E}$,
let $\lambda\_A(I)$ be the (exact) length of $I$ 
for all half-open intervals $I\in A$; 
for all open or closed intervals $I\in A$, 
respectively subtract or add $\delta$ to the length of $I$ 
to define $\lambda\_A(I)$;
let $\lambda\_A(S)=\delta$ for all singletons $S\in A$.
Extend $\lambda\_A$ to a probability measure $\mu\_A$ on $A$.


(Specifically, for 
each minimal finite union of intervals $F\in A$, 
let the connected components of $F$ be $[a\_0,b\_0),\ldots,[a\_k,b\_k)$
with $p$ elements of $\{a\_i,b\_i:i\leq k\}$ added and $q$ removed.
Partition $F$ into its atomic subsets $H\_0,\ldots,H\_n$.
Choose a positive $\mu\_A(H\_i)\in R$ for each $i$, such that
$\sum\_{i\leq n}\mu\_A(H\_i)=(p-q)\delta+\sum\_{i\leq k}(b\_i-a\_i)$.
Now extend $\mu\_A$ from the atoms to all of $A$.)


Let $U$ be a fine ultrafilter on $\mathcal{E}$.
("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$
for all $A\in\mathcal{E}$.) 
The ultraproduct measure $\mu\_U$ is $R^U$-valued and has 
the two properties you seek.


