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Clearly, this question is an application of Radon-Nikodym theorem. So, I guess all needed to show is $\mu\ll m$, but I have no clue how to show that here. The question also has hint which I struggle to understand. The hint is; let $g_A:y\to \mu(A+y)$ and define $\nu(A\times E)=\int_Eg_A(y)dm(y)$. Why do we should define such another measure? What does it give us?

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  • $\begingroup$ You have to verify that $\nu$ is countable additive on the class of finite unions of measurable rectangles. Extend it to Borel sets of $\mathbb R^{2}$ using Caratheodory's Theorem and then verify that $\mu <<m_2$. ($m_2)$ is the Lebesgue measure on $\mathbb R^{2}$). RNT applied to $\nu$ gives you the result easily. $\endgroup$
    – Kavi Rama Murthy
    Oct 28, 2019 at 23:22

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