$f_1,f_2,\dots ,$ and $f$ are in $L_{loc}^1(U)$.
I'm trying to give a counterexample where $f_n\to f$ pointwise,
but not $f_n\to f$ in $\mathcal{D}^\prime (U)$,
where $\mathcal{D}^\prime (U)$ denotes the space of all distributions on $U$.
I'm trying to come up with a sequence of functions which converges pointwise but does not converge almost everywhere or in $L^p$, and then conclude it does not converge in distribution, for convergence in distribution looks so hard to deal with. But I couldn't find anything yet. I'm not sure if this approach is reasonable.
I would appreciate any comment or hint. Thank you for reading.
This is similar to the fact that pointwise convergence does not imply $L^1$ convergence.
Construct a sequence $f_n$, each with integral $1$, that converges to $0$ pointwise.