Subspaces dimensional properties that are boundary sets of the probability measures space, defined in an infinite compactum~$X$

T. F. Zhuraev ; Q. R. Zhuvonov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 89 (2024), pp. 32-50

Résumé

In this note, we consider dimensional properties of the subspace of probability measure spaces $P(X)$ for which transfinite dimensional functions $ind$, $Ind$ и $\mathrm{dim}$ are defined. It is shown that countability of a compact set $X$ is equivalent to the existence of dimensions $ind\,P_\omega(X)$, $Ind\,P_\omega(X)$, $\mathrm{dim}\,P_\omega(X)$, $ind\,P_f(X)$, $Ind\,P_f(X)$ and $\mathrm{dim}\,P_f(X)$ for the subspaces $P_\omega(X)$, $P_f(X)$, $P_n(X)$ respectively. It is also noted that for any compact $C$-space of the subspaces $P_n(X)$, $P_\omega(X)$, $P_f(X)$ the space $P(x)$ are compact $C$-spaces. If for an infinite compact set $X$ the subspace $P_\omega(X)$ contains the Hilbert cube $\mathcal{Q}$, then there exists a number $n\in N$, $n>1$, such that $X^n \sigma^{n-1}$ contains the Hilbert cube $\mathcal{Q}$. Further, for an infinite compact set $X$, a number of subspaces $Y$ of the compact set $P (x)$ which are $\mathcal{Q}$-, $\ell_2$-, $\ell_2^f$- and $\Sigma$-manifolds are identified. In particular, for a proper closed subset $A\subset X$, the subspaces $S_p(A)$ are $\ell_2$-manifolds; for any $n \in N$, $n>$1, $P(X) \setminus P_n(X)$ are $\mathcal{Q}$-manifolds; for any proper everywhere dense countable subspace $A\subset X$, the subspace $P_\omega(A$) is the boundary set of the compact set $P (x)$. If $P_\omega(X)$ contains the Hilbert cube $\mathcal{Q}$, then the subspace $P_\omega$(X) is homeomorphic to the space $\Sigma$. It is considered in which cases everywhere dense subsets $A$ of the spaces $P(X)$ defined in an infinite compactum $X$ are its boundary set. It is also shown which everywhere dense subsets $A\subset P(X)$ and $B\subset P(Y)$ for infinite compact sets $X$ and $Y$ of the spaces $P(X)$ and $P(Y)$, respectively, are at the same time mutually homeomorphic.

Détail

Keywords: probability measures, $Ind$, boundary sets.
Mots-clés : dimensions $ind$, $\mathrm{dim}$