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To start with, I have almost no any experience in the theory of finitely additive (f.a.) measures, but I work a bit with countably additive (c.a.) ones and find the theory in the latter case amazingly beautiful. My concern is that at the moment measures have been introduced as an extension of such notions as area and volume, I can understand that the additivity property came alone naturally. However, I believe, that at that moment the choice f.a. vs. c.a. might not have any strong arguments. Later, it appeared that in many cases the space of c.a. (but not f.a.) measures is the dual of a corresponding space of all bounded continuous functions. Since the latter is a pretty "natural" object, I would say that its dual is "natural" as well. Would it be right to say that c.a. measures are more "natural" than f.a. ones, or that it appeared to be more successful/useful, and if so - why do we need the f.a. measures?

I hope, that a bit loose formulation of the question still allows for an answer.

SBF
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    The space of (signed) finitely additive measures is the dual of the space of bounded measurable functions, so the "duality argument" doesn't seem to be particularly convincing. – Martin Jun 20 '13 at 13:04
  • @Martin: that I didn't know, thanks for pointing it out. So does it mean, that for any c.a. measure there an f.a. measure such that their integrals agree on all continuous functions? – SBF Jun 20 '13 at 13:32
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    Th dual space of $\ell_\infty$ is a space of complex valued finitely additive measures denoted by $ba$. Since $\ell_\infty=\ell_1^*$, then we have natural embedding $\ell_1\hookrightarrow ba$. So the question is whether $ba=\ell_1$, or in other words what are finitely additive but not countable additive measures on $\mathbb{N}$. It turns out that question depends on axiom of choice. – Norbert Jun 20 '13 at 19:08
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    For certain axiomatizations we can have $\ell_\infty^*=\ell_1$, so there will not be "purely" finitely additive measures. Since this measures affiliated with "choice" their usage could be non-constructve. This is one of the reasons why we prefer countably additive mesures. – Norbert Jun 20 '13 at 19:09
  • Without finitely (but not countably) additive measures we would not have infinite amenable groups, which would seem like a loss, given the Wikipedia list of the properties equivalent to amenability. – ˈjuː.zɚ79365 Jun 21 '13 at 14:29

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