I posted the following question also here, but thought that I can get more answers in MO. Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, let $L_{\nu_i}:L^{1}(\nu_i)\rightarrow\mathbb R$ be the map defined by $L_{\nu_i}(g)=\int_\Omega gd\nu_i$. Denote by $[1_E]_{\nu_i}$ the equivalence class in $L^1(\nu_i)$ of the characteristic function $1_E$. Does there always exist a linear map $M:L^1(\nu_1)\rightarrow L^1(\nu_2)$ such that $$L_{\nu_2}\circ M([1_E]_{\nu_1})=L_{\nu_1}([1_E]_{\nu_1})$$ for all $E\in\Sigma$? When $\nu_1$ is absolutely continuous to $\nu_2$ then, by the RadonNykodim Theorem, the map $M$ can be given explicitly, i.e. there exists a $\Sigma$measurable function $f:\Omega\rightarrow\mathbb R$, such that $M(g):=f\cdot g$ will satisfy the equation given above. So the question becomes interesting when $\nu_1$ and $\nu_2$ are singular to each other, and the spaces $L^1(\nu_1)$ and $L^1(\nu_2)$ are not isometric. The way I think of it is, that it could characterize in some/all cases a RadonNykodim derivative when the measures are singular to each other...comments on this are welcome!
How about $M \colon L^1(\nu_1)\rightarrow L^1(\nu_2) \colon g \mapsto (x \mapsto \int g~d\nu_1)$? 

