I am trying to understand Lacey's proof of Carleson's theorem, which can be found here (https://arxiv.org/pdf/math/0307008.pdf). But I face problems even in the beginning. He is using a so called ''one sided inversion Formular'' to define the Carleson operator for all $f \in L^2(R)$, namely in the following way: \begin{align*} \mathcal{C}f (x) = \sup_{N} \left| \int_{-\infty}^N \hat{f}(\zeta) e^{i x \zeta} \mathrm{d}\zeta\right| \end{align*} where $\hat{f}$ is the Fourier transform. I don't get why the integral should exist.
Thanks in advance!