### Abstract

Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question *O* on a state η where η does not lie entirely in an eigenspace of *O* should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping *U*, with domain in the set [*Qs*τ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on *A*, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory*U* and experiment with respect to *H* is given by the statement,where *U*(*Qs*τ) is the probability measure*U* associates with the infinite instruction string *Qs*τ and ψ_{ Qsτ} is the outcome sequence obtained by carrying out *Qs*τ. *E*(*H, U*(*Qs*τ),ψ_{ Qsτ}) is the statement that all formulas in *H* with one free sequence variable which are true on Ω almost everywhere with respect to *U*(*Qs*τ) are true for ψ_{ Qsτ}. *H* is a subclass of the class of all formulas in a formal language *L*. A theorem is proved which states that, if *U*(*Qs*τ) corresponds to a nontrivial product probability measure and *U H*‐agrees with experiment, then the outcome sequence ψ_{ Qsτ} is *H*‐random. *H*‐randomness is defined here in terms of the statement *E*(*H, P*, φ). Another property of a valid physical theory, which is defined here, is that, for some *Qs*τ, *U*(*Qs*τ) must be determinable on much of *A*_{H} from ψ_{ Qsτ}. Sufficient conditions for this property to hold are given. *A*_{H} is the class of all *H* definable subsets of Ω. Some properties of the statement *E*(*H, P*, φ) are given. Among other things, it is proved that, if *E*(*H, P*, φ) holds and *P* is a nontrivial probability measure on *A*, then φ is not definable in *H*.

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