Compactification of spaces of measures and pseudocompactness

Vladimir I. Bogachev

Compactification of spaces of measures and pseudocompactness

Vladimir I. Bogachev

Abstract

We prove pseudocompactness of a Tychonoff space $X$ and the space $\mathcal{P}(X)$ of Radon probability measures on it with the weak topology under the condition that the Stone-Čech compactification of the space $\mathcal{P}(X)$ is homeomorphic to the space $\mathcal{P}(βX)$ of Radon probability measures on the Stone-Čech compactification of the space~$X$.

Compactification of spaces of measures and pseudocompactness

Abstract

We prove pseudocompactness of a Tychonoff space

and the space

of Radon probability measures on it with the weak topology under the condition that the Stone-Čech compactification of the space

is homeomorphic to the space

of Radon probability measures on the Stone-Čech compactification of the space~

Paper Structure (5 theorems, 6 equations)

This paper contains 5 theorems, 6 equations.

Key Result

Theorem 1

(i) Pseudocompactness of the space of measures $\mathcal{P}(X)$ implies pseudocompactness of $X$. (ii) The injectivity of the indicated extension of the embedding $\mathcal{P}( X)\to \mathcal{P}(\beta X)$ onto $\beta \mathcal{P}( X)$ implies pseudocompactness of both spaces $X$ and $\mathcal{P}(X)$.

Theorems & Definitions (8)

Theorem 1
Theorem 2
proof
Corollary 1
proof
Corollary 2
Proposition 1
proof