Operator Valued Measures in Quantum Mechanics: Finite and Infinite Processes

In this work, operator valued measures are used to study finite and infinite sequences of measurements. It is shown that to each such process q^Δ there is uniquely associated a probability operator measure O^qΔ which contains all the statistical properties of the process. In order to make this association for infinite processes, the operator valued equivalent of the Kolmogorov extension theorem is needed. This theorem is given and proved. It is then shown that for each q^Δ and each set E of possible outcome sequences, there are two ways to find the probability that carrying out q^Δ on a system in state ρ gives an outcome sequence φ in E. The usual method of repeating q^Δ on ρ over and over again generates a sequence α of outcome sequences φ. The probability is obtained as the limit relative frequency that α(j) is in E, for j = 0, 1, …. The other, new, method is the repeated measurement of $O_E^{\mathrm{q\Delta }}$ on ρ. The remarkable aspect of this equivalence is that the mathematical procedures of the usual method for determining if α(j) is in E or not `disappear' into the operators $O_E^{\mathrm{q\Delta }}$ of the new method. This is discussed in some detail and examples are given.