## Relations between Shy Sets and Sets of $\nu _p$-Measure Zero in Solovay's Model

### Volume 52 / 2004

Bulletin Polish Acad. Sci. Math. 52 (2004), 63-69
MSC: 28A05, 03C50, 28C20, 28D05.
DOI: 10.4064/ba52-1-7

#### Abstract

An example of a non-zero non-atomic translation-invariant Borel measure $\nu _p$ on the Banach space $\ell _p (1\le p \le \infty )$ is constructed in Solovay's model. It is established that, for $1 \le p< \infty ,$ 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.