Some questions about conditional expectations

Question

Question:

Suppose that $E｜X｜<\infty$, $\mu$ is constant. If $E(X|Y)=\mu$, then prove that $E(X|Y^3)=\mu$.

I have no idea about the relationship between $Y$ and $Y^3$. Can I say $E(X|Y)=E(X|g(Y))$?

Besides, I can figure out the expectation of X is equal to $\mu$ by using law of total expectation.

I hope someone can help me. Thank you.

Answer 1

$E(X|Y)$ is $E(X|\sigma (Y)$ by definition. Now $\sigma (Y^{3}) \subseteq \sigma (Y)$. So $E(X|Y^{3})=E(X|\sigma (Y^{3}))=E([ E(X|\sigma (Y)] |\sigma (Y^{3}))=E(\mu |\sigma (Y^{3}))=\mu$.