Example of a general random variable with finite mean but infinite variance Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, positive value, but $E(X^2)$ diverges.

One can consider a discrete random variables with support $\mathbb{N}$ such that $\mathbb{P}(X=m)=C\frac{1}{m^3}$ for some $C>0$. Then $\mathbb{E}[X]=\sum_{k\in \mathbb{N}}C\frac{1}{m^2}<\infty$ but $\mathbb{E}[X^2]=\sum_{k\in \mathbb{N}}C\frac{1}{m}$ diverges since it is the harmonic series.
