Prove that $X$ and $Z = Y^2$ are independent random variables if X, Y independent

Question

Suppose that $X$ and $Y$ are independent random variables. I wish to prove that $X$ and $Z = Y^2$ are also independent.

I know from the definition of independence that $P({X \in A}$ and $Y \in B$) = $P({X \in A})$$P({Y \in B})$, but I'm not sure how to apply this to the above problem.

Answer 1

Claim If $X$ and $Y$ are independent, then also $f(X)$ and $g(Y)$ are independent where $f,g: \mathbb{R} \to\mathbb{R}$ are measurable maps.

Proof claim: Let $A,B$ be Borel sets. Then

$$\mathbb{P}(f(X) \in A, g(Y) \in B) = \mathbb{P}(X \in f^{-1}(A), Y \in g^{-1}(B))$$ $$= \mathbb{P}(X\in f^{-1}(A)) \mathbb{P}(Y \in g^{-1}(B)) = \mathbb{P}(f(X) \in A)\mathbb{P}(g(Y) \in B)$$

and this ends the proof. $\quad \square$

Apply this with $f: x \mapsto x$ and $g: x \mapsto x^2$ to obtain that $X=f(X)$ and $g(Y) = Y^2$ are independent.