### Selections of set-valued mappings via applications

#### Abstract

Our aim is to study the problem of tightness of compact subsets of the space $M_r(X)$ of all Radon measures on the space $X$ equipped by the topology of weak convergence.

A kernel on a space $Z$ into the space $M_r(S)$ is a continuous mapping $k: Z \longrightarrow M_r(X)$.

A space $X$ is called a uniformly Prohorov space if for each $\varepsilon > 0$, any paracompact space $Z$ and any kernel $k: Z \longrightarrow M_r(X)$ there exists an upper semi-continuous compact-valued mapping $S_{(k,\varepsilon )}: Z \longrightarrow X$ such that $\mu _{(k,z)}(X \setminus S_{(k,\varepsilon )}(z)) \leq \varepsilon $

for each $z \in Z$.

Any sieve-complete space is a uniformly Prohorov space (Corollary \ref{C5.5}). Any uniformly Prohorov space is a Prohorov space.

A space $X$ is sieve-complete if and only if $X$ is an open continuous image of a paracompact \v{C}ech-complete space.

The idea of the concept of a uniformly Prohorov goes to A. Bouziad, V. Gutev and V. Valov.

A kernel on a space $Z$ into the space $M_r(S)$ is a continuous mapping $k: Z \longrightarrow M_r(X)$.

A space $X$ is called a uniformly Prohorov space if for each $\varepsilon > 0$, any paracompact space $Z$ and any kernel $k: Z \longrightarrow M_r(X)$ there exists an upper semi-continuous compact-valued mapping $S_{(k,\varepsilon )}: Z \longrightarrow X$ such that $\mu _{(k,z)}(X \setminus S_{(k,\varepsilon )}(z)) \leq \varepsilon $

for each $z \in Z$.

Any sieve-complete space is a uniformly Prohorov space (Corollary \ref{C5.5}). Any uniformly Prohorov space is a Prohorov space.

A space $X$ is sieve-complete if and only if $X$ is an open continuous image of a paracompact \v{C}ech-complete space.

The idea of the concept of a uniformly Prohorov goes to A. Bouziad, V. Gutev and V. Valov.

#### Keywords

Set-valued mapping Selection, Radon measure, support of measure, Prohorov space

#### Full Text:

PDFDOI: http://dx.doi.org/10.14510%2Flm-ns.v0i0.1384