Measuring quantities for price indices? I'm confused about how one acquires the data necessary to compute price indices--specifically, how one acquires quantity data.

Suppose I wanted to compute the Laspeyres Index, defined as

$\sum\limits_{i=1}^{N} p_{i}(t_{f}) q_{i}(t_{0}) / \sum\limits_{i=1}^{N} p_{i}(t_{0}) q_{i}(t_{0})$

where $N$ is the number of different goods, each $p_{i}$ is a price, and each $q_{i}$ is a quantity. Each $p_i$ is measured in units of currency per units of a good, and each $q_{i}$ is measured in units of a good per unit of time.

But as far as I can tell, we can't measure $q_{i}(t_{o})$ or $q_{i}(t_{f})$. These represent the instantaneous rates that a certain good is sold at a particular time, whereas data-collectors are only able to ask how much firms have sold of a good over a period of time--in other words, we can measure

$\int\limits_{t_{0}}^{t_{f}} q_{i}(t)dt$

by summing the total sales of all firms that sell good $i$ during the period between $t_{0}$ and $t_{f}$, but not

$q_{i}(t)$

for any given $t$.

How do actual econometricians measure $q_{i}(t_{o})$ and $q_{i}(t_{f})$? Do they measure the number of goods sold $x$ days after $t_{0}$ or $t_{f}$, then divide by $x$ for a sort of local approximation? Or do they use some other method?
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.