TITLE: Can a function be square integrable without being integrable?
QUESTION [11 upvotes]: Reading Tolstov's 'Fourier Series', which states that $f(x)$ is square integrable if both $f$ and its square both have finite integrals over some interval. I haven't seen this restriction on $f$ before, which makes me wonder - can squaring a function ever turn a diverging integral into a converging one?

REPLY [17 votes]: As long as the measure space is of infinite measure, this ca happen: Consider $\frac{1}{x}$ on $(1,\infty)$. 
If the measure space is finite, this cant happen by Cauchy Schwarz (with the constant 1 function as the second factor).