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!



Convergence of random variables in LP preserved under conditioning on sub sigma field
This is an immediate consequence of the [conditional Jensen inequality](http://en.wikipedia.org/wiki/Jensen%27s_inequality#Proof_3_.28general_inequality_in_a_probabilistic_setting.29).


