# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## On an Invariant Borel Measure in Hilbert Space

### Tom 52 / 2004

Bulletin Polish Acad. Sci. Math. 52 (2004), 47-51 MSC: 28A35, 28C20, 60G15. DOI: 10.4064/ba52-1-5

#### Streszczenie

An example of a nonzero $\sigma$-finite Borel measure $\mu$ with everywhere dense linear manifold ${{\Bbb I}}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space $\ell _2$ such that $\mu$ and any shift $\mu ^{(a)}$ of $\mu$ by a vector $a \in \ell _2 \setminus {{\Bbb I}}_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].