By Daniel J. Velleman

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- American Mathematical Monthly, volume 117, number 1, january 2010

**Extra resources for American Mathematical Monthly, volume 117, June July 2010**

**Sample text**

We have seen that if μ(A) = D(A, A) for a decoherence June–July 2010] FINITE QUANTUM MEASURE SPACES 515 function D, then μ is a q-measure. We will later exhibit more general q-measures that do not have this form. Simple examples of q-measures are μ(A) = ν(A)2 , where ν is a signed measure. 5) that any q-measure μ satisfies μ(∅) = 0. Example 1. Let (X, ν) be the probability space of our fair coin example. But now we have a quantum coin with q-measure μ(A) = ν(A)2 . Then the sample points have “quantum probability” 1/16 and the certain event X has “quantum probability” 1 as it should.

If μ is a q-measure we call (X, μ) a q-measure space. We have seen that if μ(A) = D(A, A) for a decoherence June–July 2010] FINITE QUANTUM MEASURE SPACES 515 function D, then μ is a q-measure. We will later exhibit more general q-measures that do not have this form. Simple examples of q-measures are μ(A) = ν(A)2 , where ν is a signed measure. 5) that any q-measure μ satisfies μ(∅) = 0. Example 1. Let (X, ν) be the probability space of our fair coin example. But now we have a quantum coin with q-measure μ(A) = ν(A)2 .

Van Gils, S. M. -O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer, New York, 1995. 6. R. D. Driver, Ordinary and Delay Differential Equations, Springer, New York, 1977. 7. G. B. , John Wiley, New York, 1999. 510 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 8. G. -O. Londen, and O. Staffans, Volterra Integral Equations and Functional Equations, Cambridge University Press, Cambridge, 1990. 9. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.