## Stochastic approximation properties in Banach spaces

### Volume 159 / 2003

Studia Mathematica 159 (2003), 103-119
MSC: Primary 46A35, 46B25, 46E30, 60B11.
DOI: 10.4064/sm159-1-5

#### Abstract

We show that a Banach space $X$ has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if $X$ has nontrivial type. If for every Radon probability on $X$, there is an operator from an $L_p$ space into $X$ whose range has probability one, then $X$ is a quotient of an $L_p$ space. This extends a theorem of Sato's which dealt with the case $p=2$. In any infinite-dimensional Banach space $X$ there is a compact set $K$ so that for any Radon probability on $X$ there is an operator range of probability one that does not contain $K$.