Abstract
This chapter introduces the notion of a cover or repertoire and its proper descriptions. Based on the new idea of relating covers and descriptions, some interesting properties of covers are defined.
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- 1.
In addition we usually may require that \(p(A)\neq 0\) for every \(A \in \alpha \).
- 2.
More exactly: for every \(B \in R(d)\) there is \(A \in \alpha \) with \(p(A\bigtriangleup B) = 0\).
- 3.
In line with our general strategy to disregard sets of probability 0, we can interpret \(A \subset d(x)\) as \(p(A \setminus d(x)) = 0\) and \(p(d(x) \setminus A) > 0\).
- 4.
In the definition of D(α), we understand a description d simply as a mapping \(d: \Omega \rightarrow \alpha \) with \(\omega \in d(\omega )\), i.e., without the additional requirement of Definition 2.3.
- 5.
If \(D(\alpha ) = \varnothing \) we obtain \({\alpha }_{c} = \varnothing \), so α c is not a cover. In this case, we add Ω to α c . To avoid this redefinition, one could define α c only for repertoires.
- 6.
- 7.
The flattening of an arbitrary cover may not exist, because \({\alpha }_{f}\) may not be a cover. An example for this is \(\alpha = R({X}^{\geq })\) for a random variable X with \(R(X) = \mathbb{R}\). In this case, α has no maximal elements. If the flattening exists, it is clearly flat. Usually we consider finite covers which guarantees that the flattening exists.
- 8.
Here we are using the countable version of Definition 2.4.
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Palm, G. (2012). Repertoires and Descriptions. In: Novelty, Information and Surprise. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29075-6_9
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DOI: https://doi.org/10.1007/978-3-642-29075-6_9
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