Originally Posted by

**Moo** If Y is g-measurable, then it's A-measurable.

Correct. But $\displaystyle Y$ is A-measurable, not g-measurable...

The inequality $\displaystyle |E[Y|g]|\leq E[|Y| |g]$ will enable to conclude by monotonicity of conditional expectation: $\displaystyle E[|Y||g]\leq E[C|g]=C$.

This inequality can be seen as an instance of the conditional Jensen inequality ($\displaystyle x\mapsto|x|$ is convex), or more elementarily as follows: denoting $\displaystyle Y_+,Y_-$ the positive and negative parts of $\displaystyle Y$,

$\displaystyle E[Y|g]=E[Y_+|g]-E[Y_-|g]\leq E[Y_+|g]\leq E[|Y||g]$

and

$\displaystyle -E[Y|g]=E[Y_-|g]-E[Y_+|g]\leq E[Y_-|g]\leq E[|Y||g]$

(each of these inequalities holds by monotonicity of conditional expetation) hence

$\displaystyle |E[Y|g]|\leq E[|Y||g]$.