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Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?

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    I'm not expert in harmonic analysis. At least, what$I$did works when the continuous functions are dense in $L^1(\mu_2)$ (yes I should precise what measure we are dealing with). I don't know whether it's the case, and under which conditions. I will look in Rudin's book.2012-09-29

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Such a condition exists. The details are in Math Overflow. To find it, we need the Radon-Nikodym theorem and approximation by trigonometric polynomials.