Example of a sequence of function $f_n$ converging pointwise to $f$ in $\Bbb R$ but not converging in $L^p(E)$.

Question

Example of a sequence of function $f_n$ converging pointwise to $f$ in $\Bbb R$ but not converging in $L^p(E)$.

Here $E$ is a measurable set.

Answer 1

Consider $$f_n=n^{\frac{1}{p}}\chi_{(0,\frac{1}{n}]}$$

Then $f_n \to 0$ pointwise. But $$\int |f_n|^p dm=1, \forall n$$