Example of an integrable function with non integrable square

Question

Let $X$ be a measure space, how can I find an example of a function $f$ such that $f \in L(X)$ - the space of real valued Lebesgue integrable functions, but $f^2 \notin L(X)$. I know this example can't be a series (with the counting measure), or an integral in $\mathbb{R}$ with the Lebesgue Measure - because in both cases, integrability means that $f$ tends to zero on infinity, which means that the square decreases faster, and it follows that the square of $f$ is integrable. Can anyone give me some hint?