Measures on a Hilbert space that are invariant with respect to shifts and orthogonal transformations

TL;DR

This paper constructs a shift and rotation-invariant measure on an infinite-dimensional Hilbert space, analogous to Lebesgue measure, and explores its properties and the structure of associated function spaces.

Contribution

It introduces a new finitely-additive measure invariant under shifts and orthogonal transformations, extending measures across different bases in an infinite-dimensional setting.

Findings

Defined a measure on measurable rectangles with convergent edge products

Decomposed the measure into mutually singular shift-invariant measures

Analyzed the structure of the space of square-integrable functions with respect to this measure

Abstract

A finitely-additive measure $\lambda $ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue measure in the sense of its invariance with respect to the above transformations. The constructed measure is defined on the ring $\cal R$ of subsets of the Hilbert space generated by measurable rectangles. A measurable rectangle is an infinite-dimensional parallelepiped such that the product of the lengths of its edges converges unconditionally. The shift and rotation-invariant measure is obtained as a continuation of a family of shift-invariant measures $\lambda _{\cal E}$, where each measure $\lambda _{\cal E}$ is defined on the ring ${\cal R}_{\cal E}$ of measurable rectangles with edges collinear to the vectors of some orthonormal basis $\cal E$ in the space $E$. An equivalence relation is introduced on the set of orthonormal bases in terms of the transition matrix from one orthonormal basis to another. The equivalence relation allows to glue measures defined on the subset rings corresponding to different bases into the one measure $\lambda $ defined on the unique ring $\cal R$. The obtained measure $\lambda $ is invariant with respect to shifts and rotations. The decomposition of the measure $\lambda$ into the sum of mutually singular shift-invariant measures is obtained. The paper describes the structure of the space $\cal H$ of numerical functions square integrable with respect to the constructed shift and rotation-invariant measure $\lambda $. The decomposition of the space $\cal H$ into the orthogonal sum of subspaces corresponding to all possible equivalence classes of bases is obtained. Unitary groups acting by means of orthogonal transformations of the argument in the space $\cal H$ of square integrable functions are investigated.