What are some interesting examples of a measure space $(X,\mu)$ such that $\mu(X)=\infty$ and
$f\in L^p (X)$ for some $1<p<\infty \Longrightarrow f\in L^1 (X)$
All the examples I have found are some what trivial, such as the point space $\left\{*\right\}$.
What I know is that, if the measure is $\sigma$-finite, there exists $f\notin L^1 (X)$ such that for all $1<p\leq\infty$, $f\in L^p (X)$. This means that no $\sigma$-finite examples exist...
Please enlighten me.