Is $ X^3 $ in $σ(X^2)? $ Is $ X^2 $ in $σ(X^3)? $

Question

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??

Answer 1

If $X$ is a random variable and $g$ a measurable function, then $\sigma(g(X))\subset \sigma(X)$. This follows immediately from definitions as if $B$ is a Borel set, then \begin{align} g(X)^{-1}(B) &= \{\omega\in\Omega : g(X)(\omega)\in B\}\\ &= \{\omega\in\Omega : X(\omega)\in g^{-1}(B)\}\\ &= X^{-1}(g^{-1}(B)). \end{align}

The converse holds as well; suppose $X$ and $Y$ are random variables such that $\sigma(Y)\subset\sigma(X)$. For each nonnegative integer $n$ let $$E_{m,n} = Y^{-1}((m2^{-n}, (m+1)2^{-n})),\quad m\in\mathbb Z. $$ Since $E_{m,n}\in \sigma(Y)$ and $\sigma(Y)\subset\sigma(X)$, we can write $E_{m,n} = X^{-1}(B_{m,n})$ for some Borel set $B_{m,n}$. Define $$f_n(x) = \sum_{m\in\mathbb Z}m2^{-n} \mathsf 1_{B_{m,n}}(x),\quad n=0,1,2,\ldots. $$ Then for each $n$ we have $$f_n(X)\leqslant Y\leqslant f_n(X)+2^{-n}. $$ It follows from monotone convergence that $f:=\lim_{n\to\infty}f_n$ exists, and $$\mathbb P\left(\lim_{n\to\infty} f_n(X)=Y \right)=1. $$

Answer 2

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 $\sigma(X^2)\subset\sigma(X^3)=\sigma(X)$.