  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.
