Geodesic
On a standard product of an arbitrary family of $\sigma$-finite Borel measures with domains in Polish spaces
Teoriâ slučajnyh processov, Tome 16 (2010) no. 1, pp. 84-93.

Voir la notice de l'article provenant de la source Math-Net.Ru

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].
Keywords: Infinite-dimensional Lebesgue measure, product of $\sigma$-finite measures. 
@article{THSP_2010_16_1_a10,
     author = {Gogi Pantsulaia},
     title = {On a standard product of an arbitrary family of $\sigma$-finite {Borel} measures with domains in {Polish} spaces},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {84--93},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2010_16_1_a10/}
}
TY  - JOUR
AU  - Gogi Pantsulaia
TI  - On a standard product of an arbitrary family of $\sigma$-finite Borel measures with domains in Polish spaces
JO  - Teoriâ slučajnyh processov
PY  - 2010
SP  - 84
EP  - 93
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2010_16_1_a10/
LA  - en
ID  - THSP_2010_16_1_a10
ER  - 
%0 Journal Article
%A Gogi Pantsulaia
%T On a standard product of an arbitrary family of $\sigma$-finite Borel measures with domains in Polish spaces
%J Teoriâ slučajnyh processov
%D 2010
%P 84-93
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2010_16_1_a10/
%G en
%F THSP_2010_16_1_a10
Gogi Pantsulaia. On a standard product of an arbitrary family of $\sigma$-finite Borel measures with domains in Polish spaces. Teoriâ slučajnyh processov, Tome 16 (2010) no. 1, pp. 84-93. http://geodesic.mathdoc.fr/item/THSP_2010_16_1_a10/

[1] R. Baker, ““Lebesgue measure” on ${\mathbb R}\sp \infty$”, Proc. Amer. Math. Soc., 113 (1991), 1023–1029

[2] R. Baker, ““Lebesgue measure” on ${\mathbb R}\sp \infty$. II”, Proc. Amer. Math. Soc., 132 (2004), 2577–2591

[3] P. R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950

[4] P. A. Loeb, D. A. Ross, “Infinite products of infinite measures”, Illinois J. Math., 49:1 (2005), 153–158

[5] J. Ma$\check{r}\acute{i}$k, “The Baire and Borel measure”, Czechoslovak Math. J., 7 (1957), 248–253

[6] G. R. Pantsulaia, Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Sci., New York, 2007

[7] G. R. Pantsulaia, “On ordinary and standard Lebesgue measures on $R^{\infty}$”, Bull. Polish Acad.Sci., 73 (2009), 209–222

[8] C. A. Rogers, Hausdorff Measures, Cambridge Univ. Press, Cambridge, 1970

[9] A. H. Stone, “Paracompactness and product spaces”, Bull. Amer. Math. Soc., 54 (1948), 977–982

[10] W. Stephen, General Topology, Addison-Wesley, London, 1970

[11] S. Y. Wen, “A certain regular property of the method I construction and packing measure”, Acta Math. Sin. (Engl. Ser.), 23:10 (2007), 1769–1776

[12] N. N. Vakhanya, V. I. Tarieladze, S. A. Chobanyan, Probability Distributions in Banach Spaces, Nauka, Moscow, 1985 (in Russian)