Measurement theory, not to be confused with (measure-theory), is the study of functions that preserve certain desirable properties. Its theoretical basis is popular in psychology, and it is related to statistical analysis of data, especially in deciding how data represents reality.
Measurement theory, not to be confused with measure-theory is the study of measurement.
Given some method for comparing objects (say by placing them next to each other to see which is the longer) one can ask what are the necessary and sufficient conditions for such a measurement procedure to give rise to a representation theorem.
Take some relation $\succeq$ and some objects $X,Y,Z\dots$. A function $f$ represents $\succeq$ when:
$$X \succeq Y \ \text{iff}\ f(X) \ge f(Y)$$
