Say we have a space $\Omega$, and a function $f:\Omega\to \mathbb R$, where $F$ is the set of all such functions $f$. Now say that we have two probability measures over the space $(\Omega, \Sigma)$: $P_1$, and $P_2$, both of which are elements of the set $\mathcal P$ of all possible probability measures on $(\Omega, \Sigma)$.
I am looking for a mapping $m:\mathcal P \times \mathcal P \times F\to \mathbb R$, that roughly captures the notion of: "a measure of the ratio/difference in expected value of $f$ under $P_1$, and $P_2$ respectively, that is invariant under linear transformations of $f$".
So a requirement is that $m(P_1,P_2, f)=m(P_1,P_2, g)$, whenever it is the case that $f(o)=a\cdot g(o) + b$ for every $o\in\Omega$ and some $a,b\in \mathbb R$.
The first guesses, $m(P_1,P_2, f)=\frac {E_{P_1}(f)}{E_{P_2}(f)}$, or $m(P_1,P_2, f)={E_{P_1}(f)}-{E_{P_2}(f)}$ are both not invariant w.r.t. linear transformations.
Is it possible to construct such a linearly-invariant "metric" $m$, (perhaps with some complex technique inside the integral that I don't know of)?