Quasi sure version of a theorem of Littlewood (Q1207659)

Let \((c_ n)_{n\in Z}\) be a sequence of complex numbers; we prove that, if \((\varepsilon_ n c_ n)_{n\in Z}\) is a Fourier-Stieltjes series for quasi all choices of signs \(\varepsilon_ n=\pm 1\), then \(\sum_{n\in Z}| c_ n|^ 2<\infty\). This improves a theorem of Littlewood and gives a topological version of a theorem of Paley and Zygmund.