Measure theoretic convergences of observables and operators

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Abstract
Definitions of different types of measure theoretic convergence for observables and operators are given. In particular we define convergence in measure, almost everywhere, everywhere, almost uniformly, and uniformly. These types of convergence are compared and characterized. Furthermore, our theory is compared to that of Segal‐Stinespring. Convergence theorems such as a bounded convergence theorem, Fatou's lemma, and a special case of Egoroff's theorem are proved. We show that the general Egoroff's theorem does not hold.

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