Menshov's set of multiplicity of measure zero

Example

Let $\mathbb{T} = \R / 2\pi\Z$. In 1916 Menshov constructed a closed set $E \subset \mathbb{T}$ of Lebesgue measure zero that is a set of multiplicity. So there is a trigonometric series $\sum_{n \in \Z} c_n e^{inx}$, with coefficients not all $0$, whose symmetric partial sums converge to $0$ at every point outside $E$.

The construction gives a nonzero positive Borel measure $\mu$ supported on a perfect null set $E$ whose Fourier–Stieltjes coefficients $\hat\mu(n) = \frac{1}{2\pi} \int e^{-inx} \, d\mu(x)$ tend to $0$ as $\abs{n} \to \infty$. The series $\sum_n \hat\mu(n) e^{inx}$ then converges to $0$ at every point outside the support of $\mu$. This follows from Riemann's principle of localization for series with coefficients tending to $0$. Its coefficients are not all $0$, since $\hat\mu(0) = \mu(\mathbb{T}) / 2\pi > 0$. Hence $E$ is not a set of uniqueness.