  We prove a de Finetti theorem for exchangeable sequences of states on test
spaces, where a test space is a generalization of the sample space of classical
probability theory and the Hilbert space of quantum theory. The standard
classical and quantum de Finetti theorems are obtained as special cases. By
working in a test space framework, the common features that are responsible for
the existence of these theorems are elucidated. In addition, the test space
framework is general enough to imply a de Finetti theorem for classical
processes. We conclude by discussing the ways in which our assumptions may
fail, leading to probabilistic models that do not have a de Finetti theorem.
