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 .

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- Nov 15th 2009, 05:26 PMKat-Mconditional expectation
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 . - Nov 15th 2009, 10:34 PMMoo
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... - Nov 16th 2009, 03:07 AMLaurent
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 ,

.