Finite Expectation Value means random variables are bounded?

Question

For a random variable $\xi $, does $|E(\xi)| < \infty $ means that there exists $M>0$, such that $|\xi | < M $ almost surely? A counter example I can think of is $\xi(x) = \frac{1}{\sqrt{x}} $, which is defined on $[0,1]$. We have $\int_0^1 \xi(x) dx = 2 $. But you can not find a uniform bound for $\xi(x)$. Is this a valid counter example?