Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If I have a sequence of uniformly integrable functions (or random variables) $X_n$ and I compose these with a function $f$ what conditions on $f$ make $f(X_n)$ uniformly integrable? Further, my intuition tells me that continuity of $f$ is not enough but I cannot think of a counter example can anyone help with with coming up with one.

share|improve this question
    
Yes, continuity is not enough, for example take an integrable $X$, which is not $L^2$, and $f(x)=x^2$. –  Davide Giraudo Nov 5 '12 at 15:57
    
But that works for affine/bounded maps, and sum of maps of this type. –  Davide Giraudo Nov 5 '12 at 16:31

1 Answer 1

In general, the class of continuous function preserving uniform integrability is the same class preserving $L^1$, namely those for which there exists $C$ such that $|f(x)| \le C|x|$ for all $x$. If you are on a finite measure space, it is enough to have this for large $|x|$, or alternatively to have constants $C$ and $D$ such that $|f(x)| \le C|x|+D$ for all $x$.

share|improve this answer
    
Can you prove this in a more formal way? Thanks! –  Adam Apr 2 '13 at 9:54

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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