# Functions of Radon Nikodym Derivatives

Suppose $$Q\ll P$$ so that $$\frac{dQ}{dP}$$ is well defined. Obviously it is true that $$\int\frac{dQ}{dP}dP = Q(\Omega) =1$$. Is it true that $$\int\left|\log\left(\frac{dQ}{dP}\right)\right|dP < \infty \quad ?$$

Is is true that $$\int \frac{dQ}{dP} dQ = \int (\frac{dQ}{dP})^2 dP < \infty$$? What about $$\int \frac{dQ}{dP} |\log(\frac{dQ}{dP})|dP$$ ? I'm trying to read about these things and explore their properties but such things aren't discussed in the books I have.

One can write $$\int|\log(\frac{dQ}{dP})|dP=\int_{1>{dQ}/{dP}>0}\log(\frac{dP}{dQ})dP+\int_{\infty>{dQ}/{dP}>1}\log(\frac{dQ}{dP})dP\\=D(P,Q|R_1)-D(P,Q|R_2)$$ where $$D$$ stands for the KL-divergence, and $$R_1$$ and $$R_2$$ are the related domains where the integral is taken. For the final result to be finite one needs Q>>P, else it is always possible to find a counterexample, where Q>>P is violated on some mesurable set and over this set the integral is infinite.