The Supremum of a Bounded Right-Continuous Process is a Random Variable

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ and $K\ge0$ be real numbers, and let $(Z_t)_{t\in[0,T]}$ be a family of random variables on $(\Omega,\mathcal{F},P)$. Let $D$ be the set of dyadic partition points of $[0,T]$, that is, the set of all numbers of the form $k\,T\,2^{-n}$ where $n$ is a natural number or zero and $k\in\{0,1,\dots,2^n\}$.

Suppose there is an event $\Omega_0\in\mathcal{F}$ with $P(\Omega_0)=1$ such that for every $\omega\in\Omega_0$ both of the following hold.

1. (Boundedness along the path.) $|Z_t(\omega)|\le K$ for every $t\in[0,T]$.

2. (Right-continuity.) The path $t\mapsto Z_t(\omega)$ is right-continuous at every $t\in[0,T)$, meaning that for every $\varepsilon>0$ there is $\eta>0$ with $|Z_s(\omega)-Z_t(\omega)|\le\varepsilon$ whenever $t\le s\le\min(t+\eta,T)$.

Write $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ elsewhere, and define

$$\overline{Z}=\sup_{t\in D}\big(|Z_t|\,\mathbf{1}_{\Omega_0}\big).$$

Then $\overline{Z}$ is a random variable with $0\le\overline{Z}\le K$, and

$$\overline{Z}(\omega)=\sup_{t\in[0,T]}|Z_t(\omega)|\qquad\text{for every }\omega\in\Omega_0 .$$