Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

Question

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ functions. However, instead of showing that $$\lim_{N\rightarrow\infty} \sum_{\left|n \right|\leq N}^\infty \hat f(n) e^{inx}= f(x) $$ holds almost everywhere for periodic functions $f \in L^2$, Lacey instead shows that for any $f\in L^2$, we have the Fourier inversion formula: $$\lim_{R\rightarrow\infty}\int_{\left|\xi\right|\leq R}\hat f(\xi) e^{2\pi i\xi x}\thinspace d\xi=f(x)$$ which holds for almost all $x\in \boldsymbol{R}$. It is not difficult to see that these two results are related, but I'm wondering how explicitly can one show that if the Fourier Inversion formula holds, then we must also have the a.e. convergence of the partial Fourier sums? Thank you in advance.