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

Understand

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.