Example of a measure space with infinite measure

Question

I'm having a little trouble solving a problem from my Real Analysis class.

The problem is from p238, "Lebesgue integration on Euclidian Space" by Frank Jones.

Q. Give an example of a measure space such that $\mu(X) = \infty,\ and\ f\in L^p(X)$ for some $1 < p < \infty \implies f\in L^1(X) $

My intuition is that this problem has to do with sigma-finite measures, but this book does not formally define sigma-finite measures in this chapter (it does on later chapters).

Would appreciate any help/hints on this problem!

Answer 1

Take $X=\{a,b\}$, with $\mu\bigl(\{a\}\bigr)=\infty$ and $\mu\bigl(\{0\}\bigr)=0$. Then\begin{align}f\in L^p(X)&\iff f(a)=0\\&\iff f\in L^1(X).\end{align}