Let $f_{n}=n^{-1/p} \chi[0,n]$ show that the sequence $(f_{n})$ converges uniformly to the $0$ function but that it does not converge in $L_{p}(\mathbb{R},B,\lambda)$
My attempts, I was trying to use the fact that the Lebesgue Dominated Convergence theorem holds if almost everywhere convergence is replaced by convergence in measure but I don't know if it is a good idea.
Note that $$ \lVert f_n \rVert_p = \int_0^n (n^{-1/p})^p \ dx = 1,$$ hence the function does not converge to zero in $L^p$. On the other hand, since $f_n \to 0$ uniformly, if $f_n \to f$ in $L^p$ this implies that $f = 0$, since by looking at every compact set $K \subseteq \mathbb{R}$ we would have $f = 0$ on $K$.