Does $f _n \to f $ pointwise imply $f _n $ converges to $f $ in $L ^p $ norm if $\{f_n\}$ is Cauchy in $L^p$?

Question

If a sequence $\{f _n \}$ of functions converges pointwise to a function $f $ does this imply that if the same sequence is a Cauchy sequence in some $L^p $ norm then it converges to the same function $f $ in the same $L^p $ norm? I believe so but how is this motivated?

Thanks in advance!

Answer 1

No, it is not true. Take $f_n$ as the indicator function of the interval $[n,n+1]$: then $f_n\to 0$ pointwise, but $\|f_n\|_p = 1$.

Even if the space $X$ has a finite measure, for instance $X=(0,1)$, by taking $f_n$ as $n^{\alpha}$ times the indicator function of $\left(1-\frac{1}{n},1\right)$ we get a counter-example.