The Entropy of Partitions

Considering a discrete probability space \((\Omega,\mathcal{P}(\Omega),\mathbb{P})\) , the following properties were required for the probability measure \(\mathbb{P}\) : A non-empty set Ω is called uncountable if there is no surjective mapping \(\mathbb{N}\to\Omega\) (in symbols: \(|\Omega|> |\mathbb{N}|\) ). It would now be obvious to demand the properties (P1)–(P3) for a probability measure \(\mathbb{P}\) even when there are uncountably many results in Ω. Unfortunately, it turns out that for uncountable Ω there are no useful mappings \(\mathbb{P}\) of this kind (see for example [Wagon85]); the uncountability of Ω severely limits the possibilities of finding a \(\mathbb{P}\) with properties (P1)–(P3). Since one cannot do without probability spaces with uncountable result sets on the one hand, and the properties indicated by (P1)–(P3) are fundamentally indispensable on the other hand, within the framework of measure theory, the definition range of \(\mathbb{P}\) (hereinafter referred to as \(\mathcal{D}\) ( \(\subseteq\mathcal{P}(\Omega)\) )) has been restricted (i.e., no longer demanding the power set of Ω) in order to thus expand the possibilities for the choice of; otherwise, the properties (P1)–(P3) should apply to instead of.