On a standard product of an arbitrary family of $\sigma$-finite Borel measures with domains in Polish spaces

Gogi Pantsulaia

Teoriâ slučajnyh processov, Tome 16 (2010) no. 1, pp. 84-93

Résumé

Let $\alpha$ be an infinite parameter set, and let $(\alpha_i)_{i \in I}$ be its any partition such that $\alpha_i$ is a non-empty finite subset for every $i \in I.$ For $j \in \alpha$, let $\mu_j $ be a $\sigma$-finite Borel measure defined on a Polish metric space $(E_j,\rho_j)$. We introduce a concept of a standard $(\alpha_i)_{i \in I}$-product of measures $(\mu_j)_{j \in \alpha}$ and investigate its some properties. As a consequence, we construct "a standard $(\alpha_i)_{i \in I}$-Lebesgue measure" on the Borel $\sigma$-algebra of subsets of $\mathbb{R}^{\alpha}$ for every infinite parameter set $\alpha$ which is invariant under a group generated by shifts. In addition, if ${\rm card}(\alpha_i)=1$ for every $i \in I$, then "a standard $(\alpha_i)_{i \in I}$-Lebesgue measure" $m^{\alpha}$ is invariant under a group generated by shifts and canonical permutations of $\mathbb{R}^{\alpha}$. As a simple consequence, we get that a "standard Lebesgue measure" $m^{\mathbb{N}}$ on $\mathbb{R^N}$ improves R. Baker's measure [2].

Détail

Keywords: Infinite-dimensional Lebesgue measure, product of $\sigma$-finite measures.