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Is anyone aware of a result which states that convergence of random variables in $\mathbb L^p$ are preserved under conditioning on sub-sigma fields?

I'm new to probability/measure theory, and trying to get a handle on the idea of combining $\mathbb L^p$ spaces with conditioning. I was trying to tackle this with martingales, but I fear that I may be way off the reservation. Any guidance would be appreciated. Thanks!

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This is an immediate consequence of the conditional Jensen inequality.

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  • $\begingroup$ Sorry, I'm being dense, but can you please be a little more explicit about the setup? I'm unclear where the convexity enters the picture. $\endgroup$ – Matt Apr 10 '14 at 4:23
  • $\begingroup$ @Matt: $\varphi(x) := |x|^p$ is a convex function. Apply the inequality, then take unconditional expectation of both sides. Finally replace $X$ by $X_n - X$. $\endgroup$ – Nate Eldredge Apr 10 '14 at 4:31

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