A very quick and simple question (i ve nobody else to ask it).
To prove that pointwise convergence does not imply convergence in $L^p$ we just simply have to take for example a sequence of functions that point wise converge to a function that is not in $L^p$ / that is unbounded in $L^p$
For example on $\omega \in (0;1)$:
$X_n(\omega ) = \frac{1}{\omega +n^{-1}}\overset{a.s.}{\rightarrow}\frac{1}{\omega }=X(\omega )$
But obviously $X_n(\omega )$ can't converge to $X(\omega )$ in $L^1(0;1)$ as $X(\omega ) = 1/\omega$ isn't even in $L^1(0;1)$