Assume $\mu \in M ( \mathbb{T} )$, space of all Radon measure over the circle. By Riesz representation theorem, we have $(C(\mathbb{T}))^{*}\simeq M(\mathbb{T})$. If $\lim_{\tau \to 0} \|\mu_{\tau}-\mu \| =0 $, prove that $\mu$ is absolutely continuous, ( absolutely continuous respect to Lebesgue measure) ?
My approach is to find some property of Fourier coefficient $\hat{\mu}(n)$ so that it is also the Fourier coefficient of some $L^1$ function $f$ , then by the uniqueness of fourier coefficient, we can obtain that $\mu = f(t) dt$,( $dt$ is Lebesgue measure). But I cannot find out what that property is. Anyone can help me answer it ?