Is a weak star convergent sequence in $L^\infty$ bounded in $L^1$ if the limit (and the sequence) are in $L^1$?

Question

I'm trying to get an $L^1$-bound for a sequence $(f_n)_n$ of which I know that:

1. $f_n \rightharpoonup^* f$ in $L^\infty(\mathbb{R})$, i.e. $\|f_n\|_\infty \leq C < \infty$.
2. All of the $f_n$ and $f$ are in $L^1(\mathbb{R})$.

Can I conclude that the norms of the $f_n$ are bounded in $L^1$?

Answer 1

Let $f_n=\chi_{(n,2n)}$ and $f=0$. Then $\int f_ng \to \int fg$ for all interagble functions $g$ but $\int f_n=n \to \infty$.