Let $I:=(0,1)$. Let $f\in L^1_{loc}(\mathbb R)$ be a periodic function with period $1$. Assume $f|_I \in L^1(I)$ and $\int_I f(x) dx=0$. Define the oscillating sequence $$ f_n(x):=f(nx). $$ This sequence is bounded in $L^1(I)$ since $\|f_n\|_{L^1(I)} = \|f\|_{L^1(I)}$ due to periodicity.
Then it is well-known that $(f_n)$ converges weakly to zero in $L^p(I)$ for all $1<p<\infty$. It converges weakly-star to zero in $L^\infty(I) = L^1(I)^*$ and in $L^1 \subset M(I) = C(\bar I)^*$.
The proofs of these statements use density of characteristic functions of intervalls (1) or of continuous functions (2) in the respective dual spaces. If $f\in L^p_{loc}(\mathbb R)$ for some $p>1$ then weak convergence in $L^1(I)$ follows from embedding.
However, these techniques do not work to show weak convergence in $L^1$, as its dual space is $L^\infty$, and these density arguments cease to work.
Another possibility is to use Dunford-Pettis theorem by showing that the sequence $(f_n)$ is uniformly integrable. But I do not see how to achieve this.
My question is: Is it possible to prove $f_n \rightharpoonup 0$ in $L^1(I)$ or is there a counterexample of a periodic function $f$, where this fails?