 In the introduction to the first paper in the series of four entitled “On Rings of Operators”, Murray and von Neumann list two reasons why they are dissatisfied with the separable Hilbert space formulation of quantum mechanics. One has to do with a property of the trace operation, which is the operation appearing in the definition of the probabilities for measurement results (the Born rule), and the other with domain problems that arise for unbounded observable operators. The trace of the identity is infinite when the separable Hilbert space is infinite-dimensional, which means that it is not possible to define a correctly normalized a priori probability for the outcome of an experiment (i.e., a measurement of an observable). By definition, the a priori probability for an experiment is that in which any two distinct outcomes are equally likely. Thus, the probability must be zero for each distinct outcome when there is an infinite number of such outcomes, which can occur if and only if the space is infinite dimensional. It is not clear why von Neumann believed that it is necessary to have an a priori probability for every experiment, especially since von Mises clearly believed that a priori probabilities (“uniform distributions” in his terminology) do not always exist (von Mises 1981, pp. 68 ff.) and von Neumann was influenced substantially by von Mises on the foundations of probability (von Neumann 1955, p. 198 fn.). Later, von Neumann changed the basis for his expressed reason for dissatisfaction with infinite dimensional Hilbert spaces from probabilistic to algebraic considerations (Birkhoff and von Neumann 1936, p. 118); namely, that it violates Hankel’s principle of the preservation of formal law, which leads one to try to preserve modularity – a condition that holds in finite-dimensional Hilbert spaces but not in infinite-dimensional Hilbert spaces. The problem with unbounded operators arises from their only being defined on a merely dense subset of the set elements of the space. This means that algebraic operations of unbounded operators (sums and products) cannot be generally defined; for example, it is possible that two unbounded operators \(A\), \(B\) are such that the range of \(B\) and the domain of \(A\) are disjoint, in which case the product \(AB\) is meaningless.