I have two questions about the uniform integrability.The definition I am using is that a...

ddcon4r
Answered
2022-07-01
I have two questions about the uniform integrability.
The definition I am using is that a class of random variables is uniformly integrable if given an , there exists a such that for any in we have .
First, is that in the definition of uniform integrability, can the density function which we compute the expectation with respect to it, vary with ?
Second, suppose I have a sequence of random variables (with varying density function w.r.t n). For the two functions , I know that for all . Does the uniform integrability of the sequence imply the uniform integrability of ?