Let $f \in L^P(T)$ for some $p>1$. If $n$th Fourier partial sum $S_n(f;t)$ converges almost everywhere as $n \rightarrow \infty$, does the limit have to be $f(t)$ almost everywhere?
I am trying to answer this question. Do you think that it would be the right thing if I start with using the fact that Cesaro means $\sigma _n(f;t) \rightarrow f(t)$ in $L_P$ sense?
If the Fourier partial sums $S_{n}(f)$ converge pointwise almost everywhere (a.e.), then the Cesaro means $\sigma_{n}(f)$ converge pointwise a.e. to the same limit. Since $\sigma_{n}(f)\rightarrow f$ in $L^{p}(\mathbb{T})$, for $1<p<\infty$, there exists a subsequence $\sigma_{n_{k}}(f)$ of the Cesaro means which converge to $f$ pointwise a.e. Therefore $\sigma_{n}(f)\rightarrow f$ pointwise a.e., whence $S_{n}(f)\rightarrow f$ pointwise a.e.
Regardless of whether we know that $S_{n}(f)$ converges pointwise a.e., it's actually true that $\sigma_{n}(f)\rightarrow f$ pointwise a.e.