Measurement theory is a branch of applied mathematics that is useful in measurement and data analysis. The fundamental idea of measurement theory is that measurements are not the same as the attribute being measured. Hence, if you want to draw conclusions about the attribute, you must take into account the nature of the correspondence between the attribute and the measurements.
Measurement theory was popularized in psychology by S. S. Stevens, who originated the idea of levels of measurement. His relevant articles include Stevens (1946, 1951, 1959, 1968).
The basic idea of any measurement theory is that a quantitative scale is a map between some empirical objects and associated numerical values. The prototype of a scale is the mapping of physical bodies to a measure of their physical mass. This mapping, however, is not arbitrary but is supposed to meet some requirements. The most important of them is that the mapping includes also a map between empirical relations between the objects to algebraic relations between the numerical values. Again the simplest example is that of physical mass. In this case the quantitative measurements are constructed in such a way that, for instance, the operation of combining objects corresponds to the addition of the masses of objects with the mass of to obtain the mass of : The physical operation of combining objects corresponds to the mathematical operation of summation. This is the most important aspect of defining a scale, since it implicitly defines the scientific meaning of the concept and determines how to use the measured values for predictions; for instance predicting the mass of a filled container from the masses of the container and that of the cargo.
There are other, more technical aspects of general measurement theory that we will not review here. They concern the types of scales and the uniqueness of scales. Scales are for instance classified as fundamental or derived, depending on whether they are based on existing scales or not. For those interested in these aspects of measurement theory we recommend the excellent summary by Suppes and Zinnes (1963).
Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971), Foundations of measurement, Vol. I: Additive and polynomial representations, New York: Academic Press.
Suppes, P., Krantz, D. H., Luce, R. D., and Tversky, A. (1989), Foundations of measurement, Vol. II: Geometrical, threshold, and probabilistic respresentations, New York: Academic Press.
Luce, R. D., Krantz, D. H., Suppes, P., and Tversky, A. (1990), Foundations of measurement, Vol. III: Representation, axiomatization, and invariance, New York: Academic Press.
Allen, M., Yen, W., (2001), Introduction to Measurement Theory, Waveland Press ISBN-13: 978-1577662303