Let $\mu$ be a probability measure on a set of $n$ elements and let $p_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by

$$ E(\mu)=-\sum_{i=1}^np_i\log(p_i) $$

with the usual convention that $0\cdot(-\infty)=0$.

The following are two fundamental properties:

Property 1:$E(\mu)$ takes its minimum on the Dirac measures.

Property 2:$E(\mu)$ takes its maximum on the uniform probability measure.

Now, for some application, I am really interested in a possible generalization when $\mu$ is a **finitely additive** probability measure on the natural numbers.

**Question:** Is it possible to define a notion of entropy of a finitely additive probability measure on the natural numbers in such a way that it verifies the following properties:

- it takes its minimum on the Dirac measures
- it takes its maximum on the finitely additive translation invariant probability measures

Any reference? Idea?

Thanks in advance,

Valerio