Show that if the Fourier series of an integrable function $f$ converges a.e., then it converges to $f$ a.e.
To solve this problem I have showed that if $F_n$ is Frejer kernel then Cesaro means $f*Fn$ of Fourier series of integrable function $f$ (i.e $L^1(S^1)$ function) converges almost everywhere to a function $g$ , also we know that $L^1$ norm $$\lim_{n\to\infty}||f*F_n-f||_1=0$$.
Now I need to show that $g$ is integrable and $L^1$ norm $||f-g||_1=0$, please help me to solve this.