I read somewhere (latest version of a webcomic to be honest) that "superposition" means:
a complex linear combination of a $0$ state and a $1$ state... Quantum mechanics is just a certain generalization of probability... Quantum mechanics has probability amplitudes, and they can be positive or negative or even complex.
Question: Does this mean that quantum wave functions are the Radon-Nikodym derivatives of (normalized) complex measures? In particular, when the complex measure has no atoms (in the sense of measure theory), then is the wave function the density of a complex-valued measure?
In particular, that would seem to imply a certain analogy between the (time-dependent) Schrodinger equation, which would then involve the time derivative of the density of a complex-valued measure, and the Fokker-Planck equation, which involves the time derivative of the density of a (real-valued) probability measure.
(Just checking Wikipedia's article about the Fokker-Planck equation now, and apparently there is a formal analogy between the two equations -- is the analogy what I've described above?)
Discussion: Wikipedia's article on probability amplitude mentions that the complex modulus of the probability amplitude is the Radon-Nikodym derivative of a probability measure, i.e. $$\int_X | \psi |^2 d\mu = 1\,, $$ but that is not my question -- what I am asking about is $\psi$ itself (i.e. not $|\psi|^2$), and whether $\psi$ is the "Radon-Nikodym derivative" of a complex measure.
The Wikipedia page for the Radon-Nikodym theorem says that the theorem generalizes even to complex-valued measures, and apprently all Hilbert spaces have the Radon-Nikodym property.
Also please note that when I say a "normalized" complex measure, I just mean a complex measure whose variation is a probability measure. So perhaps another way to state my question is:
Does the fact that $|\psi|^2$ is a Radon-Nikodym derivative of a probability measure imply that $\psi$ is the Radon-Nikodym derivative of a complex measure whose variation is the probability measure defined by $|\psi|^2$? Is it at least the Radon-Nikodym derivative of some complex measure?
Note: I decided to major in math, not physics, so although I know some basic Lagrangian and Hamiltonian mechanics, I am very ignorant of quantum mechanics, but do know a lot of relevant math for the subject (e.g. functional analysis), so I would prefer answers which more heavily emphasize mathematical formulation of the situation and not physical intuition.
Let me know if this should be migrated to Math.SE -- since what motivated the question was my reaction to an attempt to describe physical intuition, I imagine it might not be well-received there.
Related but different question: Probability amplitude in Layman's Terms