The observer, the measurement, and the observable
The observer is a physical system by itself, which possesses rulers, clocks, or any other measurement apparatus. The job of the observer and its tools is to collect information about other physical systems. We call each information possible to be collected from a system a measure. We also use the word measurement for the act of obtaining a measure.
A measure must refer to a specific characteristic of the system. For example, some curious mind could wonder about the distribution of eye colors in a system of \(n\) human beings. The colors could be brown, blue, and green, and these are the possible measures of the measurement. At the end of all measurements, \(m\) humans will have brown eyes, \(k\) humans will have blue eyes, and \(n-m-k\) humans will present green eyes. The eye color is the characteristic that has been measured by the observer, and it is called the observable. This is not the kind of example we will deal with in the classical mechanical theory but serves to illustrate the point.
We say that a measure belongs to an observable in the sense that an observer may perform a measurement on the observable, therefore collecting that measure. The set of all possible measures of a single observable is called the spectrum of the observable. In this case, a measurement is the selection of a member of the spectrum.
Here we actually find our first mathematical structure, the set theory. The spectrum of an observable is a set in the mathematical sense, and a measure is a member of the set. A measurement, therefore, is also the assignment of a member of the spectrum to a characteristic of the system. The spectrum may be limited or unlimited, countable or non-countable.