I'm reading the textbook *SURVIVAL ANALYSIS Techniques for Censored and Truncated Data* by Klein and Moeschberger, and in Chapter 3.5 it says

> Data from experiments involving right censoring can be conveniently represented by pairs of random variables $(T, C_r)$, where $C_r$ indicates whether the lifetime X is observed ($\delta = 1$) or not ($\delta = 0$), and T is equal to X if the lifetime is observed and to $C_r$ if it is right-censored, i.e., $T = \min(X, C_r)$.
Details of constructing the likelihood function for Type I censoring are as follows. For $\delta = 0$, it can be seen that
$$Pr(T, \delta = 0) = Pr(T = C_r|\delta = 0) Pr(\delta = 0) = Pr(\delta = 0)  = Pr(X > C_r) = S(C_r)$$
Also, for $\delta = 1$,
$\begin{aligned}
Pr(T,\delta = 1) &= Pr(T=X|\delta = 1)Pr(\delta = 1) \\
&= Pr (X = T | X \leq C_r) Pr (X \leq Cr) \\
&= \left(\frac{f(t)}{1-S(Cr)}\right) \left(1-S(Cr)\right) \\
&= f(t)
\end{aligned}$

I'm confused about the following:

1. when $\delta = 0$, from step 2 to step 3, it seems to let $Pr(T = C_r|\delta = 0) = 1$, but I'm not sure why is that

2. when $\delta = 1$, we have $Pr (X = T | X \leq C_r) = \frac{Pr(X = T = t \land X \leq C_r)}{Pr( X \leq C_r)}$, but why is the numerator equals to $f(t)$?

Thanks!