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.

In a countable superstable NDOP theory, the existence of a rigid ${ﬡ}_{\epsilon }$-saturated model implies the existence of $2^{\lambda }$ rigid ${ﬡ}_{\epsilon }$-saturated models of power λ for every $\lambda >2^{{ﬡ}_0}$.