How to understand $\frac{dP}{dQ}$, where $P, Q$ denote two distributions?

I am currently reading a paper named Estimating Individualized Treatment Rules Using Outcome Weighted Learning by Zhao et al., where they wrote an equation $$\frac{dP^D}{dP}=\frac{I(a=D(x))}{P(A=a)}$$ that I don't understand.

Here $P$ denotes the distribution of $(X, A, R)$. For any given deterministic function $A=D(x)$, which is a map from the space of $\mathcal{X}$ to the space of $\mathcal{A}=\{-1,1\}$, $P^D$ denotes the distribution of $(X, A, R)$ given that $A=D(X)$. Then they says, under the assumption that $P(A=a)>0$ for $a=1$ and $-1$, it is clear that $P^D$ is absolutely continuous with respect to $P$ (Why?) and ${dP^D}/{dP}={I(a=D(x))}/{P(A=a)}$.

I guess this is related to measure theory, of which I don't have much background, could anyone help explain how to understand this? This equation is important for me to understand their paper.

Any hint would be greatly appreciated!

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1 Answer

It is the Radon-Nikodym derivative (http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_derivative#Radon.E2.80.93Nikodym_derivative), which is the rate of change of one measure with respect to another.

Hope this helps.

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