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This question already has an answer here:

Given two independent random variables $X$ and $Y$ and two measurable functions $g$ and $h$, how do we show that $g(X)$ and $h(Y)$ will also be independent?

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marked as duplicate by Juho Kokkala, whuber Aug 19 '16 at 17:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.