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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 $f(x)=x^2$.2012-11-05
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    But that works for affine/bounded maps, and sum of maps of this type.2012-11-05

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