Hi, I am reading a textbook about SDE, and am very confused about the transition
$$X_T 1_{T\lt t} + E\{X_T 1_{T\geq t} | F_{t\wedge T}\}$$
$$= X_T 1_{T\lt t} + E\{X_T | F_t\} 1_{T\geq t}$$
I understand how does the sentence "However for ..." goes but I can't see how this is applied in this transition. Would greatly appreciate if someone could explain this to me.
By "the sentence you understand how goes", with $H:=E\{X_T\mid \cal F_t\}$, we have $${\color{red}{E\{X_T\mid \cal F_t\} 1_{T\ge t}}}\in \cal F_T.\tag{1}$$ So $$\begin{eqnarray*}E\{X_T1_{T\ge t}\mid \cal F_{t\wedge T}\}=\tag{by "expanding" the $\wedge$}\\ E[E\{X_T1_{T\ge t}\mid \cal F_t\}\mid \cal F_T]=\tag{since at $t$, it is known whether $T\ge t$}\\ E\{{\color{red}{E\{X_T\mid \cal F_t\} 1_{T\ge t}}}\mid \cal F_T\}=\tag{by (1)}\\ {\color{red}{E\{X_T\mid \cal F_t\} 1_{T\ge t}}},\end{eqnarray*}$$ as desired.
[Note that every line here is random, but is a deterministic function of $T$.]
