Integrable function that is not mean-square integrable

Question

Given that $X=I$, $\mathscr{F}$ is the Borel sets, and $\mu$ is Lebesgue Measure, I must show that there exists an integrable function on $X$, that is not mean-square integrable.

I know that a function that is integrable on $X$ satisfies: $$\int_X \!|f| d\mu_L < \infty $$

But I'm having trouble coming up with, or finding a process to create a function such that the following is false: $$ \int_X f^2 d \mu < \infty $$

Where do I begin?

Answer 1

Consider $x^{\alpha}$ for a suitable $\alpha$. When does $$\int_0^1 x^{\alpha} dx$$ exist?