Counterexample for weak convergence in $L^p$ if p=1

Question

For $1<p<\infty$, a sequence $(f_n)_{n \in \mathbb{N}}$ in $L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$ and $f \in L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$, I have already proven that

$f_n \longrightarrow f$ weakly as $n \longrightarrow \infty$ iff $\int_{E} f_n \,d\lambda \longrightarrow \int_E f \, d\lambda$ as $n \longrightarrow \infty$ for every measurable subset $E \subset \mathbb{R}$ of finite measure and $\sup_{n \in \mathbb{N}} ||f_n||_{L^p(\mathbb{R})}< \infty$

This statement does not hold for $p=1$, but I could not find an example. Can somebody help me?

Answer 1

David Ullrich already explained why this happens. As an example you can take $f_n=1_{[n,n+1]}$. For every measurable $E$ with finite measure we have $0\leq f_n 1_E\leq 1_E$ and $f_n 1_E\to 0$ pointwise, hence $f_n 1_E\to 0$ by the dominated convergence theorem. Yet $\int_{\mathbb{R}}f_n=1$, so the sequence cannot converge to $0$ weakly in $L^1$.