Is $ X^3 $ in $σ(X^2)? $ Is $ X^2 $ in $σ(X^3)? $ Let $X$ be a real-valued random variable that takes on both positive and negative values.we know $X^2$ and $ X^3 $ are in $ σ(X).$ Is $ X^3 $ in $σ(X^2)? $ Is $ X^2 $ in $σ(X^3)? $
I think both won't true.But, I could not see the counter example for these??  

Let $\Omega=\{\omega_1,\omega_2\},X(\omega_1)=-1,X(\omega_2)=1.$ Then $\sigma(X)=\sigma(X^3)=Pow(\Omega)$, but $\sigma(X^2)=\{\emptyset,\Omega\}.$ That's your counterexample. On the other hand $\sigma(X)=\sigma(X^3)$ for any $X$ since $x\mapsto x^3$ is bijective, so $\sigma(X^2)\subset\sigma(X^3)=\sigma(X)$.
