Extensive measurement is the simplest and most intuitive form of measurement. It applies to attributes where objects can be concatenated (combined) — placing rods end to end, combining masses on a balance — and the concatenation corresponds to addition in the numerical representation. The axioms for extensive measurement, formalized by Hölder (1901) and refined in Foundations of Measurement, provide the prototype for all other measurement structures.
Axioms
2. Associativity: (a ∘ b) ∘ c ~ a ∘ (b ∘ c)
3. Monotonicity: a ≽ b ⟺ a ∘ c ≽ b ∘ c
4. Positivity: a ∘ b ≻ a
5. Archimedean: for any a, b with a ≻ ∅, finitely many copies of a concatenated exceed b
The Representation Theorem
If these axioms are satisfied, there exists a function φ mapping objects to positive real numbers such that φ(a ∘ b) = φ(a) + φ(b) and a ≽ b ⟺ φ(a) ≥ φ(b). The representation is unique up to multiplication by a positive constant — hence extensive measurement yields a ratio scale.
Application to Psychology
Pure extensive measurement is rare in psychology because psychological attributes typically lack a physical concatenation operation. You cannot combine two loudnesses end to end. This limitation motivated the development of conjoint measurement, which achieves interval-scale measurement without concatenation. However, extensive measurement provides the conceptual foundation and mathematical template for all other measurement structures.