Assume that we work on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a filtration $\mathcal{F}_t, t\ge 0$ and stochastic basis $(\Omega, \mathcal{F}, \mathbb{P}, \mathbf{F})$ satisfies usual conditions.
Let $Z$ be a positive random variable and suppose its graph $[Z]$ is a predictable set. How to show that $Z$ is predictable time?
By predictable section theorem there exists predictable time $T_n$ such that $\mathbb{P}(T_n=\infty, Z<\infty)\le \frac{1}{2^n}$. If instead of $T_n$ we consider $S_n=T_1\wedge T_2\wedge\ldots\wedge T_n$, $S_n$ will be a decreasing sequence of stopping times. One can easily show that $$ \mathbb{P}(S_n=\infty, Z<\infty)=\mathbb{P}(T_1\wedge T_2\wedge\ldots\wedge T_n=\infty, Z<\infty)= $$ $$ =\mathbb{P}\left(\{T_1=\infty, Z<\infty\}\cap \ldots \cap\{T_n=\infty, Z<\infty\}\right)\le \mathbb{P}\left(T_n=\infty, Z<\infty\right)\le \frac{1}{2^n} $$ And defining $S(\omega)=\lim_n S_n(\omega)$ we conclude that $$ \mathbb P(S=\infty, Z<\infty)=0 $$ If we could show that
(1) the decreasing sequence of predictable times $S_n$ has a certain property: for every $\omega$ there exists $n(\omega)$ (depending on $\omega$) such that $S(\omega)=S_{n(\omega)}(\omega)$ (or, equivalently, for every $\omega$ there exists $n(\omega)$ such that $S_n(\omega)=S_{n(\omega)}(\omega)$ for all $n\ge n(\omega)$)
(2) $S=Z$ a.s.
then by (1) $S$ would be a predictable time and by (2) $Z$ must also be a predictable time.
I suspect that the relationship $S=Z$ a.s. must directly follow from $\mathbb P(S=\infty, Z<\infty)=0$, however I failed to derive it.
(1) follows from Borel-Cantelli because $\sum_n2^{-n}<\infty$. And (2) follows because $\{T_n<\infty\}\subset\{T_n=Z\}$, so a fortiori $\{S<\infty\}\subset\{S=Z\}$.