On a variant of Wiener's lemma

Let $\mu$ be a finite Borel measure on $R$. We define it Fourier transformation as follows:

$$\widehat{\mu}(\xi)=\int e^{-2\pi i\xi x}d\mu(x).$$

Fact 1: It follows from $|\widehat{\mu}(\xi+h)-\widehat{\mu}(\xi)|\le \int |e^{-2\pi i hx}-1|d\mu(x)$ that $\widehat{\mu}(\xi): R\to C $ is uniformly continuous.

Theorem (Riemann-Lebesgue) Assume that $\mu$ is absolutely continuous w.r.t. the Lebesgue measure. Then

$$\lim\limits_{|\xi|\to \infty}|\widehat{\mu}(\xi)|=0.$$

For the proof, see https://en.wikipedia.org/wiki/Riemann–Lebesgue_lemma.

Remark: The measures satisfying the above limit are called Rajchman measure. A geometric characterization of such measures can be founs in Lyons R.: Seventy years of Rajchman measures

Theorem (A variant of Wiener's lemma)   Let $\mu$ be a finite Borel measure on $R$, then 

$$\lim\limits_{T\to \infty}\frac{1}{2T}\int_{-T}^T|\widehat{\mu}(\xi)|^2d\xi=\sum_{x\in R}\mu(\{x\})^2,$$

where $\widehat{\mu}(\xi)$ is the Fourier transformation of the measure $\mu.$ 

Proof. Observe that $|\widehat{\mu}(\xi)|^2=\widehat{\mu}(\xi)\cdot \overline{\widehat{\mu}(\xi)}=\int_{R^2}e^{-2\pi i \xi (x-y)}d\mu(x)d\mu(y).$

Let $K_T(x,y)=\frac{1}{2T}\int_{-T}^Te^{-2\pi i \xi (x-y)}d\xi,$ then 

$$ K_T(x,y)= \begin{cases}
\frac{e^{2\pi iT(x-y)} - e^{-2\pi iT(x-y)}}{4Ti(x-y)} ,& x\not=y\\
1, & x=y
\end{cases} $$

Moreover, $\lim\limits_{T\to \infty}K_T(x,y)=\begin{cases}
0 ,& x\not=y\\
1, & x=y
\end{cases}$

Since $|K_T(x,y)|\le 1$, by the dominated convergence theorem we have that for all $x$,

$$\lim\limits_{T\to \infty}\int_RK_T(x,y) d\mu(y)=\mu(\{x\}).$$

Clearly, 

$$\frac{1}{2T}\int_{-T}^T|\widehat{\mu}(\xi)|^2d\xi=\int_{R^2}K_T(x,y)d\mu(x)d\mu(y).$$

It follows from Fubini's theorem,

$$\int_{R^2}K_T(x,y)d\mu(x)d\mu(y)=\int_R\mu(\{x\})d\mu(x)=\sum_{x\in R}\mu(\{x\})^2.$$

Remark: From the above result, we claim that if $\mu$ has a atom, it is not possible that $\lim\limits_{|\xi|\to \infty}\widehat{\mu}(\xi)=0,$  so $\dim_F\mu=0.$

For more information, please refer Open Quantum Systems I: The Hamiltonian Approach, LNM 1880 "Topics in Spectral Theory" by Vojkan Jaksic, Cahpter 2.

Moreover, we can ontain the following result:

$$|\mu(x)|\le \limsup\limits_{|\xi|\to \infty}|\widehat{\mu}(\xi)|.$$