Suppose we have a sequence $\{f_n\}$ of $L^1$ functions such that $||f_n||_1 \leq K_1$, then viewing $L^1(\mathbb{R}) \subset \mathcal{M}(\mathbb{R})$ where $\mathcal{M}(\mathbb{R})$ is the space of Radon measure which is isomorphic to the dual space of $C_c(\mathbb{R})$, we can extract a subsequence $\{{f_n}_k\}$which converges to a Radon measure in the weak$^*$ topology on $\mathcal{M}(\mathbb{R})$. Suppose in addition we have
$$\int_{\mathbb{R}}f_n=K_2, \forall n \in \mathbb{N}$$ and $f_n \rightarrow f$ pointwise almost everywhere in $\mathbb{R}/\{0\} $
Then can we say that $\exists C \in \mathbb{R}$ such that $$\int_{\mathbb{R}} {f_n}_k\Phi(x)dx=\int_{\mathbb{R}} f(x)\Phi(x)dx+ C\Phi(0),$$ i.e the weak$^*$ limit is of the subsequence is of the form $f+C\delta_{0}$?
Note that convergence almost everywhere in $\mathbb R\setminus\{0\}$ is equivalent to convergence almost everywhere in $\mathbb R$ since $\{0\}$ has measure zero.
Now let $f_n = n \chi_{(1,1+1/n)}$. It converges pointwise a.e. to zero, weak star to $\delta_1$.