$\mu(X)=\infty$, and $f\in L^p (X)$ for some $1

Question

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

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

Please enlighten me.