In Rudin's RCA, there is a section about trigonometric series.
Here, he defines $L^p(T)$, for $1 \le p < \infty$, to be the class of all complex, Lebesgue measurable, 2$\pi$-periodic functions on $\mathbb{R}$ for which the norm
$\Vert f \Vert_p = \Big \{ \frac{1}{2\pi} \int^{\pi}_{-\pi} \vert f(t) \vert^p \,dt \Big\}^{1/p} $
My question is focused on the periodic part. As far as I know, $L^p$ space is technically equivalence classes. So, even a function defined on $[0,2\pi]$ with $f(0) \neq f(2\pi)$ can be replaced with another function $g$ which is the same as $f$ with $g(0) = g(2\pi)$ and then it can be expanded over $\mathbb{R}$.
Or, should I understand the periodic $L^p$ function $f$ to be periodic a.e.?
Or, can say $f$ is periodic in $L^p$ if there is another function $g$ such that $f=g$ a.e. and $g$ is periodic as a function?
Any help would be appreciated!