TITLE: Entropy of a measure
QUESTION [14 upvotes]: 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

REPLY [2 votes]: I just wanted to say that topologically, your hand is forced:
Tapio's supremum definition is the way to go.
Give the space $X$ of all maps from $\mathcal{P}(\mathbb{N})$ to $[0,1]$ 
the topology of pointwise convergence.
The space $FAM$ of finitely additive probability measures on 
$\mathbb{N}$ is then a closed subspace of $X$.
Let $FSM$ be the set of finitely supported measures in $FAM$.
Let $FSA$ be the set of finite subalgebras of of $\mathcal{P}(\mathbb{N})$. 
Given $G\in FSA$ and $\mu\in FAM$, it is easy to (definably) choose $\mu_G\in FSM$ that agrees with $\mu$ on $G$: $\mu_G(\{\min(A)\})=\mu(A)$ for every atom $A\in G$. 
In $FAM$, each point $\mu$ has a neighborhood base consisting of
sets of the form $U(\mu,G,\varepsilon)$, which I use to denote
the set of $\nu\in FAM$ that agree with $\mu$ on $G$ up to error $\varepsilon$. 
Therefore, $FSM$ is dense in $FAM$.
The Shannon entropy Ent is continuous on $FSM$, and 
Ent extends uniquely to a continuous map from $FAM$ to $[0,\infty]$
given by Tapio's $Ent(\mu)=\sup\{Ent(\mu_G): G\in FSA\}$ (using my notation). 
To see this, the important step is to check that $Ent(\mu_G)\leq Ent(\mu_H)$ if $G\subseteq H$.