$f^2$ integrable and $f$ is not

Question

I'm trying to find an example of a function that is not Lebesgue integrable but $f^2$ is integrable. The problem I am trying to solve includes the converse for which i gave: $\frac{1}{\sqrt{x}}$ or the characteristic function of $(-n,n)$ as examples.

Answer 1

There are two ways for a function to be not integrable: It could be non-measurable, or the absolute value could have an infinite integral. To find examples for the latter, notice that $|f(x)|^2<|f(x)|$ whenever $0<|f(x)|<1$, so a good strategy is to look at functions with values in $(0,1)$. (An example is already given in another answer.)

If you wish to go for the non-measurability angle, the hint $(-1)^2=1$ given by John in the comments is invaluable. Can you think of a non-measurable $f$ for which $f^2=1$? I am sure I can …

Answer 2

$$f(x)=x^{-1}\mathbf 1_{[1,+\infty)}(x)$$