It states that for a measure μ on the circle, any part of μ that is not absolutely continuous with respect to the Lebesgue measure dθ can be detected by means of Fourier coefficients.
, then μ is absolutely continuous with respect to dθ.
The original statements are rather different (see Zygmund, Trigonometric Series, VII.8).
The formulation here is as in Walter Rudin, Real and Complex Analysis, p. 335.
The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H1.