Converging almost everywhere in finite space but not in p-th mean

Question

Give example of a finite measure space and a sequence of functions which converges $\mu$-a.e but not in p-mean, for any $p\geqslant 1$.

I was thinking of the following spaces $([0,1],\mathscr{B_{[0,1]}},\lambda)$ and $(\mathbb{R},\mathscr{B_{\mathbb{R}}},\lambda)$. I need to find a sequence of function that as $n\to 0$ the function goes to infinity.

I was tinking of $f(x)=x^n$ for $0<x<1$. But I am not sure.

Question:

Which function would you suggest?

Thanks in advance!

Answer 1

On the space you are considering take $f_n(x)=n$ of $0<x<\frac 1 n$ and $0$ otherwise.