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 :)

How about this one?
$$f(x)=\frac1{x(1-\ln(x))^2}$$
Then $$\int f(x)\,dx=\frac1{1-\ln x},$$
and it quickly follows that $f\in L^1$.
If $p>1$, pick $r$ with $p>r>1$ and write
$$f(x)^p=\frac1{x^r}\cdot\frac1{x^{p-r}(1-\ln(x))^{2p}},$$
note that the first factor is non-integrable and the second factor is bounded below (it is continuous and positive on $(0,1]$ and unbounded near $x=0$), so $f(x)^p$ is non-integrable.
