I have two (related), ungraded homework problems. I am hoping to receive advice on how to proceed, or to be alerted if I'm on the wrong path (or, alternatively, to have the problems solved for me).
Let $P\sim Q$ be equivalent probability measures.
QUESTION 1
Prove that $dQ/dP > 0 \,\,\,\,\,\text{ P-a.s. (and hence Q-a.s.)}$
Attempt:
$\displaystyle \ \ \int_\Omega dQ = 1 $
If $P\sim Q$ and $dQ/dP < 0$ then:
$1 = \int_\Omega \frac{dQ}{dP}dP = E^P[\frac{dQ}{dP}]\leq 0$
Which is a contradiction, therefore $\frac{dQ}{dP} > 0 \,\,\,\text{P-a.s. (and hence Q-a.s.)}$.
QUESTION 2
Prove that $dQ/dP = (dP/dQ)^{-1} \,\,\,\text{P-a.s. (and hence Q-a.s.)}$
Attempt:
By the definition of equivalent probability measures, we have that:
$\displaystyle \ \ \int_\Omega dP = \int_\Omega \frac{dP}{dQ}\frac{dQ}{dP}dP = \int_\Omega \frac{dP}{dQ}dQ$
Therefore the equality is true.