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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(b,a]$ (a,b]$ and infinitesimals to singletons
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Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that (a)

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

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

It's (a) 1) that's a problem. The Bernstein-Wattenberg construction yields a finitely-additive measure that gives (a) 1) up to infinitesimals. But it would be nice to have (a) 1) exactly.
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(b,a]$ and infinitesimals to singletons

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.