An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$ Title says most of it. Could you help me find an example?
It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would be done for all $p$. The reason I'm asking is because we proved in class that $L^p[0,1]$ is nowhere dense as a subset of $L^1[0,1]$, so there must be some $L^1[0,1]$ like this..
Thanks :)
Added: thanks for all the comments. there was some missing parts about how to use convergence theorems that i couldn't complete my own so i'd love assistance :)

$$f = \sum_n \frac{1}{2^n \cdot \int_0^1 x^{-1+1/n} \, dx} \cdot x^{-1 + 1/n}$$
EDIT: The idea is that $x^{-1 + 1/n}$ is "barely integrable", but not integrable if taken to a power $p > \frac{-1}{-1 + 1/n} \to 1$. Hence, we sum all these functions in such a way that the result is still integrable (use monotone convergence).
But because of $f \geq C_n \cdot x^{-1 + 1/n}$ for all $n$ with $C_n > 0$, it is easy to see that $f^p$ is not integrable for any $p>1$.
