The Imbedding Theorem of Kochen and Specker

Abstract

Kochen and Specker begin their analysis of the hidden variable problem by pointing out that in one sense the statistical states of a theory can always be represented as measures on a classical probability space. It is always possible to represent each physical magnitude A by a real-valued function.f _A on a space X, i.e. by a random variable on X, and associate each statistical state W with a probability measure ϱ_w on X, so that the measure of the set of points in X mapped onto the set S by the function f _A is equal to the probability assigned to the range S of A by the statistical state W, i.e.

$${\varrho _W}\left( {f_A^{ - 1}\left( S \right)} \right) = {\mu _{WA}}\left( S \right)$$

or

$$\int\limits_x {{f_A}\left( x \right)d} {\varrho _W}\left( x \right) = Ex{p_W}\left( A \right).$$