If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$.
What does it mean then to say $F_n \rightarrow F$ in $L^p(\mathbb{R})$?
(I cannot take it to mean $\langle F_n, \phi \rangle \rightarrow \langle F, \phi \rangle$ since this is just the convergence of numbers and has nothing to do with $L^p$ convergence.)