Question

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..

Answer 1

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 http://mathoverflow.net/questions/15872/

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.