($\Omega, F,P) $ is a measure space, $\ Q \ll P $ ($Q$ is related to $P$) i.e. $\ Q(A)= \int 1_ADdP $ where $\ D=dQ/dP$. Then $X$ is integrable wrt $Q$ if and only if $XD$ is integrable wrt $P$ and $\int XdQ = \int XDdP$. How do I prove this? I can only do it for $X$ a simple Borel function, not a general one.