I am looking for a non-additive, continuous set function from a simplex $\Pi_{n}$ of finite dimension $n-1$ into $[0,\infty]$. The motivation is as follows. Shannon defined the entropy $H(p_{1},\ldots,p_{n})$ of a probability measure on a finite event space. He wanted the entropy to be (i) continuous in the $p_{i}$, (ii) if all the $p_{i}$ are equal ($p_{i}=1/n$) then $H$ should be monotonic increasing in $n$ (more equally probable events mean greater entropy, so for example $H_{2}(1/2,1/2)<H_{3}(1/3,1/3,1/3)$), and (iii) if choices are subdivided the entropy should remain constant, such that, for example, $H_{3}(p_{1},p_{2},p_{3})= H_{2}(p_{1},q)+qH_{2}(p_{2}/q,p_{3}/q)$ for $q=p_{2}+p_{3}$.
The only function fulfilling these requirements is the Shannon entropy
$$ H(p)=-K\sum_{i=1}^{n}p_{i}\log{}p_{i}\mbox{ ($\clubsuit$)} $$
I want to generalize the Shannon entropy from elements of $\Pi_{n}$ to subsets of $\Pi_{n}$ (e.g. Borel subsets). Let $\pi$ be a subset of $\Pi_{n}$. I guess here are some commonsense requirements for $\eta$, the generalization of $H$. (i) $\eta(\{p\})=H(p)$. (ii) $\eta$ is continuous in the sense that
$$ \mbox{If }\bigcap_{k=1}^{\infty}\pi_{k}=\pi\mbox{ with }\pi_{k}\subseteq\pi_{k-1}\mbox{ then }\eta(\pi)=\lim_{k\rightarrow\infty}\eta(\pi_{k}) $$
and
$$ \mbox{If }\bigcup_{k=1}^{\infty}\pi_{k}=\pi\mbox{ with }\pi_{k}\supseteq\pi_{k-1}\mbox{ then }\eta(\pi)=\lim_{k\rightarrow\infty}\eta(\pi_{k}) $$
One requirement that I can't quite articulate yet is that $\eta$ won't be additive but more of an averaging function, such that $\eta(\{p,q\})$ is somewhere between $H(p)$ and $H(q)$, certainly not the sum of them. The intuition is that if a rational agent can permissibly hold these two probability functions, her information is roughly the average of the information where she is in the situation that she can hold only one of them.
My question for you: is it possible to give an explicit formula for $\eta$ as in ($\clubsuit$)? I imagine it would look very similar to ($\clubsuit$) and involve some kind of integral over the points of the simplex.