Möbius Transform of a Continuous Possibility Function

In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.

There is a transformation for discrete possibility functions which is based on sorting the possibility values. I couldn't find a continuous counterpart to the transformation. Are Möbius functions inherently discrete?

Definition of Möbius transform from "G. Klir, Uncertainty and Information: Foundations of Generalized Information Theory, Hoboken N.J.: Wiley-Interscience, 2006":

Every set function

$\mu: \mathcal{P}(X) \rightarrow R$

where X is a finite set, can be uniquely represented by another set function

$^{\mu}m: \mathcal{P}(X) \rightarrow R$

via the formula

$^{\mu}m(A) = \sum_{B | B \subseteq A} (-1)^{|A-B|} \mu(B)$

for all $A \in \mathcal{P}(X)$. This formula is called Möbius transform and function $^{\mu}m(A)$ is called a Möbius representation of $\mu$.

(Another reference: http://en.wikipedia.org/wiki/Fuzzy_measure_theory)

Edit: A possibility function is a function defined on elements of $X$ and assign a number between 0 and 1 to each member:

$r: X \rightarrow [0,1], \quad \max_{x \in X} r(x) = 1$

To measure possibility of subsets of $X$, the possibility measure is defined on $\mathcal{C}$, a family of subsets of $X$ with a good algebraic structure (e.g. a $\sigma$-algebra):

• $\operatorname{Pos}: \mathcal{C} \rightarrow [0,1]$
• $\operatorname{Pos}(\emptyset) = 0,$
• $\operatorname{Pos}(X) = 1,$
• $\forall A,B \in \mathcal{C} : \operatorname{Pos}(A \cup B) = \max (\operatorname{Pos}(A), \operatorname{Pos}(B))$

This is the set function which I want to transform. With this definition, we can define $\operatorname{Pos}$ based on possibility function:

$\forall A \in \mathcal{C}: \operatorname{Pos}(A) = \sup_{x \in A}(r(x))$

In order to transform the possibility measure $\operatorname{Pos}$, $P(X)$ in the definition of $\mu$ must be changed so that: $\mu: \mathcal{C} \rightarrow [0,1]$.

My problem: In case $X=[0,1]$, the subsets in $\mathcal{C}$ are not countable. So, the summation, and $|A-B|$ in the definition of the transformation are meaningless!

In conclusion, I want $X$ to be uncountable (but with finite bounds).

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I don't think functional-analysis is the right tag, but I am not sure what it should be replaced with –  Yemon Choi Jan 4 '12 at 16:09
This question is in the context of Possibility and Evidence theories. The set function μ is the general form of the possibility and belief measures in these theories. I couldn't find these theories among tags. But added another two tags which are relevant. –  SadeghD Jan 4 '12 at 18:12
(Not set-theory.) –  Andres Caicedo Jan 4 '12 at 22:03

(edit) Okay, so as far as i can see you want to find a replacement for the mobius transform, but for a $\sigma$-algebra. In fact I'm going to guess that your $\sigma$-algebra is the measurable sets in the unit interval, based on what you've said.