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