I have $\mu$ a complex borel measure on $\mathbb(R)$, whose Fourier transform goes to zero as $ \ksi$ goes to $ \infty$. I need to prove that $ |\mu|$(singleton) = 0. Should I approach $\mu$ by a sequence of absolutely continuous (to lebesgue measure) measures in the weak star topology, or is there a simpler way to do this?

I have $\mu$ a complex borel measure on $\mathbb(R)$, whose Fourier transform goes to zero as $ \xi$ goes to $ \infty$. I need to prove that $ |\mu|$(singleton) = 0. Should I approach $\mu$ by a sequence of absolutely continuous (to lebesgue measure) measures in the weak star topology, or is there a simpler way to do this?

I have $\mu$ a complex borel measure, whose Fourier transform goes to zero as $ \ksi$ goes to $ \infty$. I need to prove that $ |\mu|$(singleton) = 0. Should I approach $\mu$ by a sequence of absolutely continuous (to lebesgue measure) measures in the weak star topology, or is there a simpler way to do this?

I have $\mu$ a complex borel measure on $\mathbb(R)$, whose Fourier transform goes to zero as $ \ksi$ goes to $ \infty$. I need to prove that $ |\mu|$(singleton) = 0. Should I approach $\mu$ by a sequence of absolutely continuous (to lebesgue measure) measures in the weak star topology, or is there a simpler way to do this?