Re: decomposition of r.v.

Hello,

I don't quite understand your question... You're quoting decompositions of martingales, but is X,Y or Z a martingale ?

If not, your question is trivial, since a F-measurable rv will be G-measurable : consider Z=X and Y= 0.

Re: decomposition of r.v.

aren't martingales ( and are no filtrations).

Sorry, I've forgotten to mention, that are not measurable with respect to .

Thank you.

Re: decomposition of r.v.

Okay, here is a guess (I didn't entirely check if it works)

Consider to be the set of all elements belonging to and not belonging to . It can be proved that is a sigma-algebra. Hence we can consider the conditional expectation of X with respect .

So if you take and , it should work.