Let Y be a positive or integrable randoma variable on the probability space( ).
If |Y| C, show that |E(Y| )| C, where is a sub algebra of .
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
.