It is well known that Entropy is additive, and that it is the only sensible choice for measuring uncertainty if we want additivity to hold, i.e.
$H(XY) = H(X)+H(Y)$
or more explicitly, if we have the constraint that
$\int_\chi\int_\chi p_1(x)p_2(x) f(p_1(x)p_2(x)) dx^2 = \int_\chi p_1(x) f(p_1(x)) dx + \int_\chi p_2(x) f(p_2(x))dx$
with $\int_\chi p_1 dx = \int_\chi p_2 dx = 1$. The only choice is $f(u) = k \log(u)$.
There is a similar problem concerning f-divergences, where we can introduce an additivity constraint:
$\int_\chi\int_\chi p_1(x)p_2(x) f\left(\frac{p_1(x)p_2(x)}{q_1(x)q_2(x)}\right) dx^2 = \int_\chi p_1(x) f\left(\frac{p_1(x)}{q_1(x)}\right) dx + \int_\chi p_2(x) f\left(\frac{p_2(x)}{q_2(x)}\right)dx$
I am pretty sure that the only valid functions $f$ are
$f(u) = (A/u+B)\log(u)$
but I do not know how to show that (or even if) this exhausts all the possibilities.
EDIT:
Corrected u to 1/u, to fit the definition of $u = p/q$ (usually it is $q/p$).
How I get to $(\frac{A}{u}+B)\log(u)$
First of all, if $f(u)$ and $g(u)$ are solutions, so is their linear combination: $a f(u) + b g(u)$. I can find two solutions that are not linearly related
1) $f(u) = k_1 \log u$
2) $f(u) = k_2 \frac{\log u}{u}$ (this effectively switches p and q)
these can be shown to work, making all the solutions of the form I proposed valid as they are linear combinations of (1) and (2). The problem I have is that I am not sure that there are not other functions that are not linearly related to either (1) or (2).