A sequence converging uniformly to $0$ on $\RR$ but not in $L^1(\RR)$

Uniform convergence on $\RR$, which implies pointwise convergence and convergence almost everywhere, does not imply convergence in $L^1(\RR)$. For $n\geq 1$ let \[ f_n \coloneqq \frac{1}{n} \chi_{[0, n]}\colon\RR\to\RR. \] Then $\norm{f_n}_\infty = \frac1n\to 0$, so $f_n\to 0$ uniformly on $\RR$. But $\int_\RR \abs{f_n - 0}\dx = \frac1n\cdot n = 1$ for every $n$, so $f_n\not\to 0$ in $L^1(\RR)$.

Give a sequence that converges uniformly, pointwise, and a.e., but not in $L^1$.

What links to this

$L^1$