Rajchman measure

Remarks

On the circle $\mathbb{T} = \R / 2\pi\Z$, the dual is $\Z$, and the condition reads $\hat\mu(n) = \int e^{-int} \, d\mu(t) \to 0$ as $\abs{n} \to \infty$.

Example: let $d\mu = f \, dt$ with $f \in L^1(\mathbb{T})$. Then $\hat\mu(n)$ is the $n$-th Fourier coefficient of $f$ up to a constant factor, and it tends to $0$ by the Riemann–Lebesgue lemma. So every absolutely continuous measure is Rajchman.

Non-example: the point mass $\delta_0$ has $\hat\delta_0(n) = 1$ for every $n$, so it is not Rajchman. The Cantor measure on the middle-thirds Cantor set, viewed on $\R/\Z$, is not Rajchman either. It is invariant under $x \mapsto 3x \bmod 1$, so its coefficients at $n = 3^k$ all equal its coefficient at $n = 1$, which is not $0$.

The Rajchman measures form a closed ideal of the measure algebra $M(G)$ that contains $L^1(G)$. Menshov showed that there are singular measures on the circle that are Rajchman.