The observer, the measurement, and the observable
The observer\label{word: 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\label{word: measure}. We also use the word measurement\label{word: measurement} for the act of collecting a measure.
A measure must refer to a certain 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\label{word: 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\label{word: spectrum} of the observable. In this case, a measurement is the selection of a member of the spectrum.
Measurement as the interaction of the observer with the system
Statistical issues
Type of Observables (intensive and extensive)?
Observable dependent and independent measurements
The question of consciousness consciousness and reality