TITLE: Wiener Corollary in "An introduction to harmonic analysis" by Yitzhak Katznelson
QUESTION [6 upvotes]: I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson  which is stated as follows:
Corollary. Let $\mu\in M(\mathbb T)$. Then
$$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$
It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not.
What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

(note added by YC: the result is stated without proof at the end of Chapter I Section 7.11 in the 1976 Dover edition)

REPLY [7 votes]: As far as I understand, $\mu$ is a Borel probability measure on the unit circle $\mathbb{T}$. Then
$$
\frac{1}{2N+1}\sum\limits_{n=-N}^N|\hat\mu(n)|^2=\int\int\frac1{2N+1}\sum_{n=-N}^N (x/y)^nd\mu(x)d\mu(y).
$$
The integrand
$$
\frac1{2N+1}\sum_{n=-N}^N (x/y)^n
$$
has absolute value at most 1 and converges pointwise to 1 for $x=y$ and to $0$ for $x\ne y$ (the sum is a geometric progression; you may sum it up to see that it is bounded when $x\ne y$). Thus, by the Dominated Convergence Theorem, the limit is the $\mu\times 
\mu$-measure of the diagonal $\{x=y\}$, which is exactly the LHS.