Let $\eta_t$ be a Markov process defined in a probability space $(\Omega, F, P)$ and taking values in $[0,1]$.
Call $\rho=\inf\{t\geq 0 :\eta_t\geq 1/2\}$ and $S=\inf\{t\geq \rho: \eta_t=0\}$.
Let $A\in F$ be a measurable set such that $P(A)>0$ and $$P(\rho<\infty|A)=1.$$ Calling $F_t=\sigma(\eta_s, s\in [0, t])$ the sigma algebra generated by the process up to time $t$, is that true the following equality?
\begin{align}P(S<\infty|A)&=E(\mathbb I_{\{S<\infty\}}|A)\\ &=E(E(I_{\{S<\infty\}}|F_{\rho}, \rho<+\infty, A)|A)\\ &=E(I_{\{S<\infty\}}|F_{\rho}, \rho<+\infty, A)E(1|A)\\ &=E(I_{\{S<\infty\}}|F_{\rho}, \rho<+\infty, A)\\ &=P(S<\infty|F_{\rho}, \rho<+\infty, A). \end{align} I guess there is an error in my argument. Where I am wrong?