1. ## conditional expectation

Let Y be a positive or integrable randoma variable on the probability space($\displaystyle \Omega$ $\displaystyle A$ $\displaystyle P$).
If |Y| $\displaystyle \leq$ C, show that |E(Y|$\displaystyle g$)|$\displaystyle \leq$ C, where $\displaystyle g$ is a sub $\displaystyle \sigma$ algebra of $\displaystyle A$.

2. 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...

3. 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
$\displaystyle |E[Y|g]|\leq E[|Y||g]$.