If $X$ and $Y$ are independent, then $X$ and $W$ are independent

Question

Let $X$ and $Y$ be independent random variables. Suppose $W$ is a random variable such that $W=Y$ almost surely. Are $X$ and $W$ independent?

Initial Work: Define $A=\{ \omega \; | \; Y(\omega) \ne W(\omega) \}$. Then, $P(A)=0$ and $P(A)=1$ since $W=Y$ a.s.

Answer 1

\begin{align} \mathsf{P}(X\in A,W\in B)&=\mathsf{P}(X\in A,W\in B,W=Y)+\mathsf{P}(X\in A,W\in B,W\ne Y) \\ &=\mathsf{P}(X\in A,Y\in B)+0 \\ &=\mathsf{P}(X\in A)\mathsf{P}(Y\in B)=\mathsf{P}(X\in A)\mathsf{P}(W\in B). \end{align}