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An example of a non-zero non-atomic translation-invariant Borel measure ${\nu }_p$ on the Banach space ${\ell }_p\left(1\le p\le \infty \right)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "${\nu }_p$-almost every element of ${\ell }_p$ has a property P" implies that “almost every” element of ${\ell }_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold ${}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift ${\mu }^{\left(a\right)}$ of μ by a vector $a\in \ell ₂{\setminus }_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].