Today (because my players were abusing the discretisation) I briefly looked back into a thing I call "synthetic measure theory". It's the beautiful union of functional programming and functional analysis, where you represent probability distributions by their integration operator
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Note that for noncommutative algebras A there's still a useful theory of integrals ∫: A -> C. They're called "states" or more generally "weights" or less generally "traces": https://en.wikipedia.org/wiki/Von_Neumann_algebra#Weights,_states,_and_traces … This is what people often use in noncommutative integration theory.
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The theory of vector-valued measures is well understood (in this case, projection-valued measures, projections on closed subspaces are a subspace of bounded operators). For an existence theorem, see https://almostsure.wordpress.com/2010/03/10/existence-of-the-stochastic-integral-2-vector-valued-measures/ … for instance.
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You may be able to use this theorem in that link using an algebra of polynomials to show the spectral theorem for bounded normal operators. For unbounded, you need more work.
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