Let $X,Y: (\Omega, \mathcal F, \mathbb P)\to E$ be independent and $A \in \mathcal F$. Are $X_{\restriction A}, Y_{\restriction A}$ still independent? Let $X,Y: (\Omega, \mathcal F, \mathbb P) \to E$ be independent random variables. Let $A \in \mathcal F$ be an event and $\mathcal A := \{F \cap A \mid F \in \mathcal F\}$ the sub-$\sigma$-algebra induced by $A$, ans assume $\mathbb{P}[A]>0$. Let $$\mathbb{P}|_A(\cdot)=\frac{\mathbb{P}(\cdot\cap A)}{\mathbb{P}(A)}$$
Is it true that $X_{\restriction A}, Y_{\restriction A}:(A, \mathcal A, \mathbb P_{\restriction \mathcal A}) \to E$ are still independent?
If $A \in \sigma(X) \cap \sigma(Y)$, then the answerer is clearly yes. Could you elaborate on the general case?

What you are asking about is equivalent to asking whether $X$ and $Y$ are conditionally independent given $A$. Let $E = \mathbb{R}$ and pick $c \in \mathbb{R}$. Let $A = \{X + Y = c\}$. On $A$ we have $X = c - Y$, so, unless $P(A) = 0$, the restrictions of $X$ and $Y$ to $A$ are not independent.
