Hello,
If Y is g-measurable, then it's A-measurable.
And thus E(h(Y)|g)=h(Y), for any g-measurable function h...
Correct. But is A-measurable, not g-measurable...
The inequality will enable to conclude by monotonicity of conditional expectation: .
This inequality can be seen as an instance of the conditional Jensen inequality ( is convex), or more elementarily as follows: denoting the positive and negative parts of ,
and
(each of these inequalities holds by monotonicity of conditional expetation) hence
.