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measure-theory proof-verification

I think that your proof is fine. Regarding the missing inequality, you have already shown that $\frac{d\nu}{d\mu}\frac{d\mu}{d\nu}=1$ a.e., so of course you have $$ 0<\frac{d\nu}{d\mu}\frac{d\mu}{d\nu}<\infty. $$

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