On convergence in $L^p$

Question

I was trying to understand an exercise about convergence in $L^p$. I was asked to investigate the punctual convergence and the $L^p(\Bbb R)$ convergence, $1\le p\le +\infty$, of $u_n(x)=1/n*e^{-|x|/n}$, $x \in \Bbb R$. I studied $\lim_{n\to +\infty} f_n(x) = 0$ and I proved that $f_n(x)$ belongs to $L^p(\Bbb R)$. Isn't that enough for the $L^p(\Bbb R)$ convergence? Or do I need Beppo-Levi's theorem or Lebesgue's Theorem? Thanks.

Answer 1

It is in theory not sufficient to establish the pointwise convergence and that each $f_n$ is in $L^p$, as the example $f_n(x)=n^{1/p}\mathbf 1_{(0,1/n)}$ shows.

Here one have to compute the $L^p$ norm of $u_n$ for a fixed $n$, by doing the substitution $t=x/n$.

Answer 2

For $1 < p <\infty$ we have $\int |u_n(x)|^{p}dx=\int \frac 1 {n^{p}} e^{-p|x|/n} dx=n^{1-p}\int e^{-p|y|} dy \to 0 $, so $u_n \to 0$ in $L^{p}$. The same compuation shows that convergence fails when $p=1$. Also convergence for $p=\infty$ is obvious since $0\leq u_n(x) \leq \frac 1 n$.

Pointwise convergence does not imply convergence in $L^{p}$ as this example with $p=1$ shows.