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!


This is an immediate consequence of the conditional Jensen inequality.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.