I am reading Protter's book "Stochastic Integration and Differential Equations". He (page 51) defines $H$ to be a simple predictable processes if it has a representation $$H_t=H_0\mathbf1_{\{0\}}+\sum_{i=1}^nH_i\mathbf1_{(T_i,T_{i+1}]}(t)$$ where $0=T_1\le \dots\le T_n<\infty$ are stopping times and $H_i\in\mathcal{F}_{T_i}$ with $|H_i|<\infty$ a.s. The collection of all such processes is denoted with S.
And then he imposes the uniform convergence (in $(t,\omega)$) topology on the space S. What makes me confused is that according to this definition, it seems that the uniform metric is not really well-defined as it can be $\infty$.
What am I missing here?