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.

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.

Reviewer: Sergey Ludkovsky (Moskva)

##### MSC:

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 |