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 space into X whose range has probability one, then X is a quotient of an 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-1-5,
author = {V. P. Fonf and W. B. Johnson and G. Pisier and D. Preiss},
title = {Stochastic approximation properties in Banach spaces},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {103-119},
zbl = {1059.46017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-1-5}
}
V. P. Fonf; W. B. Johnson; G. Pisier; D. Preiss. Stochastic approximation properties in Banach spaces. Studia Mathematica, Tome 157 (2003) pp. 103-119. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-1-5/