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.
