I am a student majoring applied mathematics. While writing my thesis paper, I found out that I need to prove or disprove the following statement:
Suppose that $P$ and $Q$ are probability measures on the same measurable space, and $Q$ is absolutely continuous with respect to $P$. Then, the range of $dQ/dP$ is a measurable subset of $\mathbb{R}$.
I made several attempts to prove the statement or find a counterexample, but I am having difficulties. Is there anyone who knows an answer to this question? Thank you.
The question is not quite meaningful. The Radon-Nikodym derivative $dQ/dP$ can be any nonnegative (a.e.) member $f$ of $L^1(P)$ with $\int f\; dP = 1$. However, members of $L^1$ are not really functions, but rather equivalence classes of functions under equality almost everywhere. So there really is nothing you can point to as the range of $f$. You can change $f$ arbitrarily on a set of $P$-measure $0$, and it is the same as a member of $L^1(P)$, but it has a different range. Now given any non-measurable subset $Y$ of $\mathbb R$ and any subset $X$ of $\mathbb R$ with cardinality $c$ and measure $0$, there is a function taking $X$ onto $Y$. So it can be that some representatives of $f$ have measurable range while others do not.