In an exercise I found here (3.11), given a Poisson point process $\eta$ with intensity $\lambda$ it's required to
Show that $$Cov (\eta(B_1), \eta(B_2)) = \lambda( B_1 \cap B_2)$$ where $B_1, B_2 \in \chi$
I can't understand how to prove it: I know that by definition of covariance I have $$Cov(\eta(B_1), \eta(B_2)) = E[\eta(B_1) \eta(B_2) ] - E[\eta(B_1)]E[ \eta(B_2)]$$
and since it's a poisson point process I have $E[\eta(B_1)] = \lambda(B_1)$ and $E[\eta(B_2)] = \lambda(B_2)$, but I can't understand what to do with the other expectation...
Hint: $$\eta(B_1)\eta(B_2)=(\eta(B_1\setminus B_2)+\eta (B_2 \cap B_1))(\eta(B_2\setminus B_1)+\eta (B_1 \cap B_2)). $$ Use the facts that $\eta (A)$ and $\eta (B)$ are independent if $A \cap B$ is empty and $E\eta(B_1 \cap B_2)^{2}=\lambda (B_1\cap B_2)$.