Consider $f$ being a measurable function on $R^n$ such that $\int_{E} e^{|f|}=1$ ($E$ measurable) and $f$ vanishes outside $E$ . Then $f\in L^p(R^n)$ for all $p\in (0,\infty)$.
I tried using that measure of $E$ cannot be bigger than $1$ and the formulae
$\int|f|^p=p\int_0^\infty \alpha^{p-1}\omega(\alpha)d\alpha$
where $\omega(\alpha)=\{x \in R^n: |f(x)|>\alpha\}$.