Measuring quantities for price indices?
The notation $q_{i}(t_{0})$ does not necessarily mean number of goods sold in instant $t_0$. It can also mean the number of goods sold in the period denoted by $t_0$. As you point out the first one would be somewhat difficult to measure.

In fact I have not yet seen the notation you use. The notation I am familiar with is usually something like $q_{i,t_0}$, which better indicates that time is treated as a discrete variable. The unit of time could be anything, but it is usually a year or quarter.