How to prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$

Question

Assume that P and Q are equivalent probability measures on a measurable space $(\Omega , \mathcal{F})$ and for a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$. Prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$.

Thank you!

Answer 1

You can use the fact that the density of $Q$ with respect to $P$ is (strictly) positive since $Q$ and $P$ are equivalent, and that $\mathcal{G}$ contains at least one set that is not a $P$-null set.

Answer 2

Because $\frac{dQ}{dP}$ is a density it has to hold $$E_P\left[\frac{dQ}{dP}\right] = 1$$

Now assume $$Y := E_P(\frac{dQ}{dP}|\mathcal{G}) = 0$$ and you would get the contradiction $$ 1 = E_P\left[\frac{dQ}{dP}\right] = E_P\left[Y\right] = E[0] = 0$$