Let be a measureable space (in fact Polish, so we can guarantee the existence of regular conditional probabilities), two probability measures on and two sub -algebras.

Let and be the regular conditional probabilities of and given .

If on and -almost everywhere

then

(i think it should say -almost everywhere too, but maybe it is implicit)

Any help regarding this step? What would be a proper approach?

I have tried to use the next properties:

-Conditional expectation (with respect to )

Just in case means the radon-nikodym derivative on

Any help would be greatly appreciated!