My professor has mentioned in class that if we have two $2\pi$-periodic functions $f$ and $g$ that are both in $L_1(\mathbb{T})$, then
$(f*g)(t) := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(s-t)g(s)ds$ is only guaranteed to be defined for almost every $t$.
I thought I followed it at the time, but now that I'm sitting at home looking at the remark, I just don't see it. If we took $f_{1}$ and $g_{1}$ such that $f=f_{1}$ almost everywhere and $g = g_{1}$ almost everywhere, wouldnt we have
$\frac{1}{2\pi}\int_{-\pi}^{\pi}f_{1}(s-t)g_{1}(s)ds = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(s-t)g(s)ds$ ?
... or was I missing his point completely? I thought it had something to do with the fact that functions in the $L_{1}(\mathbb{T})$ are actually equivalence classes.