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?
• 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. Oct 28, 2019 at 23:22