Linear operators with compact supports, probability measures and Milyutin maps

Abstract

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for 0-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and κ-metrizability, are preserved under Milyutin maps.

Functional extenders and set-valued retractions

“…For any functional µ: C * (X) → R we define its support S(µ) to be the following subset of the Čech-Stone compactification βX of X (see [8] for a similar definition):…”

“…In this section we describe the supports of the functionals from sets R max (X) and R min (X). For any functional µ : C * (X) → R we define its support S(µ) to be the following subset of the Čech-Stone compactification βX of X (see [14] for a similar definition): Definition 2.1. S(µ) is the set of all x ∈ βX such that for every neighborhood O x of x in βX there exist f, g ∈ C * (X) with βf |(βX\O x ) = βg|(βX\O x ) and µ(f ) = µ(g).…”

On a Theorem of Arvanitakis

“…The theory of maps between compact spaces admitting averaging operators was developed by Pelczyński [13]. For noncompact spaces we use the following definition [17]: a surjective continuous map f : X → Y admits an averaging operator with compact supports if there exists an embedding g : Y → P β (X) such that supp(g(y)) ⊂ f −1 (y) for all y ∈ Y . Then the regular linear operator u :…”