# conditional expectation

• November 15th 2009, 05:26 PM
Kat-M
conditional expectation
Let Y be a positive or integrable randoma variable on the probability space( $\Omega$ $A$ $P$).
If |Y| $\leq$ C, show that |E(Y| $g$)| $\leq$ C, where $g$ is a sub $\sigma$ algebra of $A$.
• November 15th 2009, 10:34 PM
Moo
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...
• November 16th 2009, 03:07 AM
Laurent
Quote:

Originally Posted by Moo
If Y is g-measurable, then it's A-measurable.

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

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

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

$E[Y|g]=E[Y_+|g]-E[Y_-|g]\leq E[Y_+|g]\leq E[|Y||g]$
and
$-E[Y|g]=E[Y_-|g]-E[Y_+|g]\leq E[Y_-|g]\leq E[|Y||g]$
(each of these inequalities holds by monotonicity of conditional expetation) hence
$|E[Y|g]|\leq E[|Y||g]$.