Uniform integrability under continuous functions

AI Thread Summary
The discussion centers on whether the composition of a uniform integrable function X with a continuous function g results in a uniform integrable function g(X). One participant expresses skepticism about this claim and seeks counterexamples. Another contributor notes that the statement holds true if the support of g is compact or if the range of the sequence {X_n} is compact. They suggest that the question may be better suited for a more specialized forum focused on measure theory. The conversation highlights the nuances of uniform integrability in the context of continuous functions.
jk_zhengli
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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.
 
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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.
 
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