I am reading the book Applied Analysis by John K. Hunter. And Lemma 7.8 states
Suppose that $f\in H^k(T)$ for $k>1/2$. Let $$S_N(x) = \frac{1}{\sqrt{2\pi}}\sum_{n=-N}^N \hat{f_n}e^{inx}$$ be the Nth partial sum of the Fourier series of f, and define $$||f^{(k)}|| = (\sum_{n\in \mathbb{Z}}|n|^{2k}|\hat{f_n}|^2)^{1/2}$$ Then there is a constant $C_k$, independent of $f$, such that $$||S_N-f||_{\infty} \leq \frac{C_k}{N^{k-1/2}}||f^{(k)}||$$ and $(S_N)$ converges uniformly to $f$ as $N\rightarrow \infty$
Here $H^k(T) = \{f\in L^2(T)|f(x) = \sum_{n\in\mathbb{Z}}c_n e^{inx},\sum_{n\in\mathbb{Z}}|n|^{2k}|\hat{f_n}|^2<+\infty\}$ and $T = [0,2\pi]$.
My question is is this uniform convergence up tp a zero measure? If I change the value of $f$ on a zero measure set, the above is still true. But the value of $||S_N-f||_{\infty}$ will be affected.
If $f \in H^{k}(T)$ then $f$ is actually continuous and the essential supremum is same as actual supremum: $\sum n^{2k}|c_n|^{2}<\infty$ implies $\sum|c_n| \leq (\sum n^{2k}|c_n|^{2})^{1/2}(\sum \frac 1 {n^{2k}})^{1/2}$ and $\sum \frac 1 {n^{2k}}<\infty$. But $\sum |c_n| <\infty$ implies that the series $\sum c_ne^{inx}$ conveges uniformly, so $f$ is continuous.