example of simple random variable on [0,1] that has finite mean but infinite variance

Question

Could anybody give an example of simple, nonnegative random variable, i.e. $$X = \sum_{i=1}^n a_i 1_{A_i}$$ where $a_i \ge 0, A_i \in \mathscr F, A_i \cap A_j = \emptyset$ on $\Omega = [0,1], \mathscr F = \mathscr B[0,1], \mathbb P = Leb$ such that $$E(X) < \infty, E(X^2) = +\infty$$

Here

Answer 1

All moments of bounded random variables are finite. If $|X| \leq M < \infty$ is finite, then $E[|X|^n] \leq M^n$ which is also finite.

Answer 2

So do you mean $[0,1]$ is the sample space? Then take $$ X(t) = \frac{1}{\sqrt{t}} $$ so that $E[X]= 2$ and $E[X^2] = +\infty$.