Constructing and combining predictable stopping times

Show that if $S_{1}, \ldots , S_{n}$ are predictable stopping times, then so are $S_{1} \wedge \cdots \wedge S_{n}$ and $S_{1} \vee \cdots \vee S_{n}$.

If $A_{t}$ is a predictable process with paths that are right continuous with left limits and $a>0$, show $T=\inf \left\{ t>0: \Delta A_{t}>a\right\}$ is a predictable stopping time.

Show that if $T$ is a predictable stopping time, $B \in \mathcal{F}_{T_{-}}$, and $T_{B}(\omega )$ is defined to be equal to $T(\omega )$ if $\omega \in B$ and equal to $\infty$ otherwise, then $T_{B}$ is a predictable stopping time.

Let $A$ be a predictable increasing process and let $S_{k}$ be the $k$ th time $A$ jumps more than $\varepsilon$. Thus $S_{0}=0$, a.s., and $S_{k+1}=\inf \left\{ t>S_{k}: \Delta A_{t}>\varepsilon \right\}$. Show each $S_{k}$ is a predictable stopping time.

Show that the stopping times $T_{n}$ defined in the proof of Proposition 16.23 are predictable.

From that proof: $R_{1}, R_{2}, \ldots$ is a countable collection of predictable stopping times (constructed from a decomposition of certain other predictable stopping times $U_{m}^{i}$). Define $T_{1}=R_{1}$; define $T_{2}$ by setting $T_{2}(\omega )=R_{2}(\omega )$ if $R_{2}(\omega ) \neq R_{1}(\omega )$ and $T_2(\omega ) = \infty$ otherwise; and in general set $T_{n}(\omega )=R_{n}(\omega )$ if $R_{n}(\omega ) \neq R_{1}(\omega ), \ldots , R_{n-1}(\omega )$ and $T_{n}(\omega )=\infty$ otherwise.