Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence of i.i.d. random variables with common distribution function $\mu$. Consider a property $A$, such that $\mu(A)>0$. Define $T$ to be stopping time associated with event $A$. $$T=\min \{n: X_n\in A\}$$ The question is: Is $X_T$ independent of $T$?
I have two arguments: one concludes that they are independent and the other concludes they aren't.
Not independent: Lets say $T=6$ for some realization. Now if I didn't tell you that $T=6$, your distribution for $X_6$ would be $\mu$ i.e. $P(X_6\in B)=\mu(B)$. On the other hand if I told you that $T=6$, then your distribution is a conditional one. With information about the value of $T$, $P(X_6\in B)=\frac{\mu(A\cap B)}{\mu(A)}$. Hence $T$ and $X_T$ are not independent.
Independent: It doesn't matter what the value of $T$ is. $$P(X_T\in B\mid T=1)=P(X_T\in B\mid T=2)=\cdots$$ This argument makes it seem that $T$ and $X_T$ are independent.
Which argument is wrong and why? I'm confused.