EN
An example of a non-zero non-atomic translation-invariant Borel measure $ν_p$ on the Banach space $ℓ_p (1 ≤ p ≤ ∞)$ is constructed in Solovay's model. It is established that, for 1 ≤ p < ∞, the condition "$ν_p$-almost every element of $ℓ_p$ has a property P" implies that "almost every" element of $ℓ_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.