On a new measure on infinite dimensional unite cube

I. Sh. Jabbarov

Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 122-133.

Résumé

Measure Theory plays an important role in many questions of Mathematics. The notion of a measure being introduced as a generalization of a notion of the size of a segment made many of limiting processes be a formal procedure, and by this reason stood very productive in the questions of Harmonic analysis. Discovery of Haar measure was a valuable event for the harmonic analysis in topological groups. It stood clear that many of measures, particularly, the product of Lebesgue measure in finite dimensional cube $[0,1]^{n} $ could be considered as a Haar measure. The product measure has many important properties concerning projections (see [1,3]). The theorems of Fubini and Tonelly made it very useful in applications. In this work we show that the coinsidence of considered measures, observed in finite dimensional case, impossible for infinite dimensional case, despite that such a representation was in use without proof. Considering infinite dimensional unite cube $\Omega =[0,1]\times [0,1]\times \cdots $, we define in this cube the Tichonoff metric by a special way despite that it induces the same topology. This makes possible to introduce a regular measure eliminating difficulties connected with concentration of a measure, with the progress of a dimension, around the bound. We use the metric to define a set function in the algebra of open balls defining their measure as a volume of open balls. By this way we introduce a new measure in infinite dimensional unite cube different from the Haar and product measures and discuss some differences between introduced measure and the product measure. Main difference between the introduced measure and Haar measure consisted in non invariance of the first. The difference between the new measure and product measure connected with the property: let we are given with a infinite family of open balls every of which does not contain any other with total finite measure; then they have an empty intersection. Consequently, every point contained in by a finite number of considered balls only. This property does not satisfied by cylindrical set. For example, let $D_{1} =I_{1} \times I\times I\times \cdots $, $D_{2} =I_{2} \times I_{1} \times I\times I\times \cdots,\dots$ $$ I=[0,1], I_{k} =\left[0,\frac{k}{k+1} \right], k=1,2,.... $$ It clear that every of these cylindrical sets does contain any other, but their intersection is not empty (contains zero). This makes two measures currently different. Bibliography: 8 titles.

Détail

Keywords: Measure theory, Lebesgue measure, Haar measure, Borel measure.

Bibliographie

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