zbMATH — the first resource for mathematics

Spaces and maps of idempotent measures. (English) Zbl 1220.18002
Izv. Math. 74, No. 3, 481-499 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 3, 45-64 (2010).
Idempotent measures and their spaces are considered for the category \({\mathcal C}omp\) of compact Hausdorff spaces \(X\) and continuous maps. Maps \(\odot : {\mathbb R}\times C(X)\ni (c,x)\mapsto c_X+\phi \in C(X)\) and \(\oplus : C(X)\times C(X)\ni (\phi ,\psi )\mapsto \max \{ \phi , \psi \} \in C(X)\) of idempotent mathematics are used, where \(C(X)=C(X,{\mathbb R})\) denotes the Banach space of continuous real-valued functions on \(X\) equipped with the sup-norm, \(c_X\) denotes the constant map having the value \(c\). An idempotent measure (or Maslov measure) is a functional \(\mu : C(X)\to \mathbb R\) satisfying the conditions: (1) \(\mu (c_X)=c\); (2) \(\mu (c\odot \phi ) = c \odot \mu (\phi )\); (3) \(\mu (\phi \oplus \psi ) = \mu (\phi ) \oplus \mu (\psi )\).
The set \(I(X)\) of all idempotent measures on \(X\) is equipped with the weak \(*\) topology. It is demonstrated that the functor \(I\) preserves the class of embeddings, the class of surjective maps, intersections for any closed subsets of \(X\), pre-images, the functor \(I\) is normal. Section 3 of the paper shows that this functor defines a monad \(\mathbf I\) in the category \({\mathcal C}omp\).
The author proves that the hyperspace monad is a submonad of the monad \(\mathbf I\). Then idempotent Milyutin maps \(f: X\to Y\) are investigated, which are defined satisfying conditions of theorem 4.1: if \(X\) is a compact metrizable space, then there exists a zero-dimensional compact metrizable space \(X\) and continuous maps \(f: X\to Y\) and \(s: Y\to I(X)\) such that \(supp (s(y)) \subset f^{-1}(y)\) for each \(y\in Y\). Theorem 4.2 states that the idempotent measure functor is open. The author also proves that the correspondence assigning to any pair of idempotent measures the set of measures on their product which have the given marginals is not continuous. Finally examples and open questions are discussed in the article.

18B30 Categories of topological spaces and continuous mappings (MSC2010)
60B05 Probability measures on topological spaces
54B20 Hyperspaces in general topology
12K10 Semifields
16Y60 Semirings
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
Full Text: DOI