How important results in classical statistics be generalized when codomain of the measure is not $\mathbb R$?
From Can expectation be calculated if the events aren't numbered?, I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc.
My guess is that the formulae for the expected value and the variance don't change: $\operatorname{E}[X] = \frac{\int f(x)}{∫dx}$ and $\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]$. The classical Gauss distribution stems from the classical central limit theorem: continuously applying the convolution operation on a function, then you will get the Gauss distribution at the end. So I guess the generalized central limit theorem is the Cauchy theorem, and the Gauss distribution is the universal point that all sequences converge at?
As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
Note that the probability to for an event to happen is still in $[0, 1]$.
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