Uniform integrability under continuous functions

[jk_zhengli]

Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI?

I don't think g(X) is UI, but I have trouble finding counter examples.

Thanks.

 

[Protege_de_GS]

This statement is true whenever \supp g (You can prove this with Heine Borel) or the range of {X_n} is compact.
Since now you have the compactness relaxed, you can pursue that direction.

Also, the foundation of this question is more towards Intro to Meas. Theory, you may consider re-post in the right domain.