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Let $X$ be a countable set and $\mathcal M(X)$ be the set of finitely additive probability measures on $X$. If $\mu\in\mathcal M(X)$, I define the entropy of $\mu$ to be

$$ E(\mu)=\sup\left\{\left.-\sum\mu(A_i)\log(\mu(A_i))\\ \right|\\ X=\bigcup A_i, A_i \text{ pairwise disjoint}\right\} $$

If $\mu$ is a Dirac measure, then $E(\mu)=0$. I am looking for non trivial examples for which this happens.

Question: Does there exist a finitely additive (non countably additive) probability measure with zero entropy?

P.s. using AC, one can ''construct'' an example taking the measure defined by a free ultrafilter on $X$. But... I would prefer to avoid the use of AC.

P.p.s. I could ask the same question also for $E(\mu)=\infty$, since the unique examples that I have in mind make use of the existence of an invariant measure on an amenable group, which follows by Hahn-Banach.

I have the feeling that the problem is deeper: it is difficult to prove the existence of a finitely additive non countably additive probability measure without using some version of AC. I am just asking if there is any particular case..

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Without AC, you are not even able to show that any purely finitely additive probability measure exists: econ.kuleuven.be/ces/discussionpapers/Dps10/DPS1010.pdf –  Michael Greinecker Oct 17 '11 at 13:31
    
indeed I suspect that. It is not clear if the paper is published somewhere.. –  Valerio Capraro Oct 17 '11 at 13:42
    
No, it isn't published yet. –  Michael Greinecker Oct 17 '11 at 13:48
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The first question is equivalent to the existence of a non-principal ultrafilter: a measure has entropy 0 if and only if every set has measure 0 or 1. This question was asked (and answered) at mathoverflow.net/questions/15872/non-principal-ultrafilters-on The second question is equivalent to the existence of a strictly finitely additive measure. Given any strictly finitely additive measure $\mu$, you can take its average with the countably additive measure $\nu(n)=C/(n\log n)^2$ to get a measure with infinite entropy. –  Anthony Quas Oct 17 '11 at 13:54
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Indeed, it is consistent with ZF that every finitely additive measure on a sigma-algebra is countably additive. –  Gerald Edgar Oct 17 '11 at 13:58
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up vote 2 down vote accepted

Posted above as a comment, but re-posted here as an answer following suggestion of @Benoit Kloeckner so original q does not appear unanswered.

The first question is equivalent to the existence of a non-principal ultrafilter: a measure has entropy 0 if and only if every set has measure 0 or 1. This question was asked (and answered) at Non-principal ultrafilters on ω

The second question is equivalent to the existence of a strictly finitely additive measure. Given any strictly finitely additive measure $\mu$, you can take its average with the countably additive measure $\nu(n)=C/(n\log n)^2$ to get a measure with infinite entropy.

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thank you, Anthony. –  Valerio Capraro Oct 17 '11 at 19:29
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