Are their examples where hölders inequality can't be applied to just measurable functions i.e. if $f$ and $g$ are measurable and $fg$ integrable, but $f$ may be not in $L^p$ or $g$ may be not in $L^q$ ($\frac{1}{q}+\frac{1}{p}=1$) can I still conclude that
$\|fg\|_1\leq \|f\|_p \|g\|_q$ ?
If neither $f$, nor $g$ are zero almost everywhere, then $\Vert f\Vert_p, \Vert g\Vert_q>0$. Hence, if $f\notin L^p$ or $g\notin L^q$, then that just means $\Vert f\Vert_p=\infty$, respectively $\Vert g\Vert_q=\infty$ and Hölder's inequality just says $\Vert fg\Vert_1<\infty =\Vert f\Vert_p \Vert g\Vert_q$. So, Hölder holds true, but does not tell us anything interesting.
If $f$ is zero almost everywhere and $g\notin L^q$, then the problem is that the RHS of Hölder's inequality is "$0\cdot \infty$" and so we would need to make sense of that first.
The reason why we typically assume $f\in L^p, g\in L^q$ is to make sure that $fg\in L^1$. We do not want to assume $fg\in L^1$ (in a way that is "Hölder inequality light").