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Alexander Pruss
Alexander Pruss
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(ba,a]$b]$ and infinitesimals to singletons

Corrected typo, tried to improve formatting, added NSA tag
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Douglas Zare
Douglas Zare
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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) 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?

  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 (a1) that's a problem. The Bernstein-Wattenberg construction yields a finitely-additive measure that gives (a1) up to infinitesimals. But it would be nice to have (a1) exactly.

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.

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.

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Alexander Pruss
Alexander Pruss
  • 2.6k
  • 14
  • 28

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.