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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.

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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

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 $x$.

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

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