Prove that $E[E[Y \mid X] \cdot E[Y \mid Z]] = (E[Y])^2$ if $X$ and $Z$ are independent

Question

$X,Y,Z$ are random variables. $Y$ is integrable. $X$ and $Z$ are independent, so the espectation of $X$ doesn't affect the expectation of $Z$.

$\sigma(\cdot)$ is the $\sigma$-algebra generated by the respective random variable. This is my resoning:

$$ \sigma(E[Y\mid X]) \subset \sigma(X) $$ $$ \sigma(E[Y\mid Z]) \subset \sigma(Z) $$

Then $\sigma(E[Y\mid X])$ and $\sigma(E[Y\mid Z])$ are independient too:

$$ E(E[Y\mid X]\cdot E[Y\mid Z]) = E(E[Y\mid X])\cdot E(E[Y\mid Z]) = E[Y]\cdot E[Y] = (E[Y])^2 $$

I want to know if I'm right.

Answer 1

Your calculations are correct.