Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says:
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the part where Bjork seems to imply that
$E^Q (X/G) . E^P (L/G) = E^P[(L.E^Q(X/G))/G]$
is valid since $E(x.y/\tau) = yE(x/\tau)$ if $y$ is $\tau$ measurable.
However in the next step, Bjork seems to say
$E^P[(L.E^Q(X/G))/G] = L.E^Q(X/G)$
Why would this be valid?
Moreover the RHS seems to imply
$E^P[(L.X)/G] = L.X$
Why is this valid?