For definiteness, let's just consider $L^1$ functions. Then any $L^1$ function can be considered as a classical or tempered distribution. In particular, suppose we have a sequence of $L^1$ function $f_n$ and an $L^1$ function $f$ s.t. $f_n \to f$ as distributions (either classical or tempered). In other words, $$\int (f_n - f)\phi \to 0,$$ for all $\phi \in C_c^\infty(\Omega)$ for some open $\Omega \subseteq \mathbb R^n$ (for classical distributions) or for all $\phi \in \mathcal{S}(\mathbb R^n)$ for tempered distributions.
What can we say about other forms of convergence $f_n \to f$, e.g. a.e. pointwise, in measure, in $L^p$, or having a subsequence such that one of those holds? It's easy to see that in general, we can't have convergence in $L^1$, since e.g. $f_n = 1_{[n,n+1]}$ converges to $0$ in the above sense, but not in $L^1$.