Related Measures

Question

($\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.

Answer 1

• Let $X$ integrable wrt $Q$. Write $X=X^+-X^-$, where $X^+$ and $X^-$ are the positive and negative parts of $X$. Let $X_n$ a sequence of step functions which converge pointwise to $X^+$ and $X_n\leq X^+$. We can show by Fatou's lemma that $$\int X^+DdP=\int \liminf_n X_nDdP\leq \liminf_n\int X_nDdP=\liminf_n\int X_ndQ\leq \int X_n^+dQ,$$ hence $DX^+$ is $P$ integrable. Do the same for $X^-$ to get the result. Use the same idea to get the converse, using the fact that $D$ is non-negative.
• You got $\int XdQ=\int XDdP$ for each $X$ simple. By a monotone convergence argument, you have to show that this equality holds if $X\geq 0$ and is integrable. Then you will deduce the general case.