I posted the following question also [here](https://math.stackexchange.com/questions/225297/linear-maps-between-the-l1-spaces-of-two-singular-measures), 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 Radon-Nykodim 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 Radon-Nykodim derivative when the measures are singular to each other...comments on this are welcome!



Linear Maps between $L^1$-spaces of singular measures
How about $M \colon L^1(\nu\_1)\rightarrow L^1(\nu\_2) \colon g \mapsto (x \mapsto \int g~d\nu\_1)$?


