Stopping Time of a Filtration on a Compact Time Interval

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be a real number, and let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$.

A stopping time of $(\mathcal{F}_t)_{t\in[0,T]}$ is a function $\tau:\Omega\to[0,T]$ such that

$$\{\omega\in\Omega:\ \tau(\omega)\le t\}\in\mathcal{F}_t\qquad\text{for every }t\in[0,T].$$